On connected components of Shimura varieties

Thomas J. Haines

Abstract

We study the cohomology of connected components of Shimura vari-eties SKp coming from the group GSp

2g, by an approach modeled on thestabilization of the twisted trace formula, due to Kottwitz and Shelstad.More precisely, for each character ω on the group of connected compo-nents of SKp we define an operator L(ω) on the cohomology groups withcompact supports Hi

c(SKp , Q`), and then we prove that the virtual traceof the composition of L(ω) with a Hecke operator f away from p and asufficiently high power of a geometric Frobenius Φr

p, can be expressed asa sum of ω-weighted (twisted) orbital integrals (where ω-weighted meansthat the orbital integrals and twisted orbital integrals occuring here eachhave a weighting factor coming from the character ω). As the crucial step,we define and study a new invariant α1(γ0; γ, δ) which is a refinement ofthe invariant α(γ0; γ, δ) defined by Kottwitz. This is done by using atheorem of Reimann and Zink.

Introduction

A Shimura variety SK is constructed from the data of a reductive group G overQ, a G(R)-conjugacy class X of R-group homomorphisms C× → GR, and acompact open subgroup K of G(Af ), subject to certain conditions [1]. The fieldof definition of a Shimura variety is a number field, called the reflex (or Shimura)field, denoted here by E. The relevance of Shimura varieties to automorphicforms stems from the following basic facts:

1) the etale cohomology groups H ic(SK ⊗ Q, Q`) admit an action of the Hecke

algebra H = Cc(K\G(Af )/K) ofK-bi-invariant compactly supported Q`-valuedfunctions on G(Af ) (here ` is a fixed rational prime), which commutes with thenatural action of the Galois group Gal(Q/E); and2) the Lefschetz number of this action contains important arithmetic infor-mation which can be related (via the trace formula) to certain automorphicrepresentations of the group G.

A Shimura variety is (usually) disconnected, and each connected componentis defined over a finite extension of the reflex field. If one forgets the Heckealgebra action and considers only the action of the Galois group, no extra in-formation is gained by studying the individual connected components of the

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Shimura variety. However, if one considers the simultaneous actions of the Ga-lois group and the Hecke algebra, some extra information is encoded in the thestory for the individual connected components. The goal of this paper is toprove a formula, in the setting of the individual connected components of theShimura varieties associated to the group of symplectic similitudes over Q, forthe Lefschetz number of a Hecke operator times a high power of Frobenius atprimes where the variety and the Hecke operator both have good reduction.The ultimate goal, to be pursued in a future work, is to use this formula toextract the extra information alluded to above, which should result in a betterunderstanding of the way the individual connected components contribute tothe hypothetical correspondence between certain autormorphic representationsof the group of symplectic similitudes and certain Galois representations. Partof this understanding should include the description of the cohomology groupsof the individual connected components in terms of automorphic representationsof various groups. Since the individual connected components are themselvesalgebraic varieties defined over number fields, such a description is expected dueto general conjectures of Langlands.

To be more precise, let (G,X,K) be a PEL-type Shimura datum and letSK denote the corresponding Shimura variety over the reflex field E. Supposep is a prime number such that GQp

is unramified, suppose Kp ⊂ G(Qp) is ahyperspecial maximal compact open subgroup and Kp ⊂ G(Ap

f ) is a sufficientlysmall compact open subgroup; further suppose K = KpKp, a compact opensubgroup of G(Af ). Then SK has good reduction at any prime p of E dividingp. Choose a prime ` different from p and consider the alternating sum of the etalecohomology groups with compact support H•

c (SK⊗Q , Q`) =∑

(−1)iHic(SK⊗

Q , Q`) as a virtual Q`[Gal(Q/E)] × H-module, where H = Cc(G(Af )//K) isthe Hecke algebra. According to Langlands’ conjectures, it should be possibleto understand this virtual module in terms of automorphic representations ofcertain groups. As a first step towards this goal one needs to find a convenientformula for the Lefschetz number

Tr(Φrp f ; H•

c (SK⊗Q , Q`)).

Here Φp denotes the inverse of an arithmetic Frobenius over p, which acts onthe cohomology groups by transport of structure; f = f pfp ∈ H is spherical atp, and r is a sufficiently high integer (depending on f). In the case where SK isof PEL-type and G is of type A or C, Kottwitz [11] proved that the Lefschetznumber takes the form

∑

(γ0;γ,δ)

c(γ0; γ, δ) Oγ(fp) TOδ(φr),

where the sum is over those equivalence classes of group-theoretic triples (γ0; γ, δ)such that the Kottwitz invariant α(γ0; γ, δ) = 1, where c(γ0; γ, δ) is a volumeterm, Oγ(fp) is the orbital integral of fp and TOδ(φr) is the twisted orbitalintegral of some function φr on G(Qpr ) depending on r and the cocharacter µdetermined by the datum X.

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This expression bears a resemblance to the geometric side of the Arthur-Selberg trace formula. In fact it is possible, assuming some conjectures in har-monic analysis on reductive groups, to rewrite it as a linear combination of theelliptic contributions to the stable trace formulas for certain functions fH , whereH ranges over all the endoscopic groups for G (see [9]). Then one can use theArthur-Selberg trace formula for the groups H to give an explicit conjecturaldescription of the virtual Q`[Gal(Q/E)]×H-module H•

c (SK⊗Q , Q`) in terms ofautomorphic representations for the group G and the λ-adic representations thatconjecturally realize the Langlands correspondence for G (see §7 and §10 of [9]).For certain Shimura varieties, this description has been completely proved: forShimura varieties attached to GSp4, in a work of Laumon [19] (Kottwitz’ work[9] being made unconditional by the resolution of the problems in harmonic anal-ysis due to Hales [4] and Waldspurger [33], [34], or Weissauer-Schroder [35]); forShimura varieties attached to “fake” unitary groups (for central division algebrasover CM fields) by Kottwitz [10]; for Picard modular surfaces [17]. By taking fto be the trivial Hecke operator in these examples, one can derive an expressionfor the local zeta function of SK at p in terms of automorphic L-functions of G,as predicted by Langlands (see [14], [15], [16]). This extends some of the earlierwork of Shimura on the zeta functions of modular and (quaternion) nonmodularcurves (see [29], [30] and [31], and note that Shimura dealt in his works withmany Shimura varieties which are not of PEL type), as well as that of Ihara [5]and others. It is also related to more recent work of Reimann [26].

Now let G be the group of symplectic similitudes GSp2g. It is well-knownthat the Shimura variety SK is geometrically disconnected. Each connectedcomponent is a quasi-projective variety over some number field. In this pa-per we are concerned with the analogue of the Lefschetz number above foreach connected component of SK . To this end we introduce an operator L(ω)on Hi

c(SK ⊗Q , Q`) as follows (see §2): we give the set of connected compo-nents the structure of a finite abelian group πp (noncanonically) and consider

a character ω : πp → Q×` . Let Xa denote the component of SK indexed by

a ∈ πp. Then we define L(ω) to act on the direct summand H ic(Xa⊗Q , Q`) of

Hic(SK⊗Q , Q`) by the scalar ω(a) ∈ Q

×` . The purpose of introducing the op-

erators L(ω) is that a suitable linear combination of them gives the projectionHi

c(SK ⊗Q , Q`) → Hic(Xa⊗Q , Q`) and thus we can isolate the cohomology

of any connected component. The main theorem of this paper is the followingformula for the “twisted” Lefschetz number (Theorem 8.2).

Theorem 0.1. Let p > 2 be a prime number. Let (GSp2g, X,K) be a Shimuradatum for which Kp is a hyperspecial maximal compact subgroup of GSp2g(Qp),and let SK denote the corresponding Shimura variety. Let ω be a character ofthe group of connected components πp of SK . Let f ∈ H be the Hecke operatorcoming from an element g ∈ G(Ap

f ) (cf. §2). Then for all sufficiently highintegers r (depending on f),

Tr(Φrp f L(ω) ; H•

c (SK⊗Q , Q`))

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is equal to an expression of the form

∑

(γ0;γ,δ)

c(γ0; γ, δ) ωp(p−1) 〈α1(γ0; γ, δ) , κ0〉 Oω

γ (fp) TOωδ (φr).

where we take the sum over all G-equivalence classes of triples (γ0; γ, δ) suchthat

1. α(γ0; γ, δ) = 1,

2. γ0 is ω-special,

and where κ0 ∈ Z(I1)Γ is any element which satisfies ∂(κ0) = a.

The notation is too complicated to explain completely here (see §5 and §6for definitions, and Theorem 8.2 for a more precise statement), so we contentourselves with a brief description of how the formula is related to that of Kot-twitz above. The sum is taken over the ω-special triples (γ0; γ, δ) which havetrivial Kottwitz invariant. The notion of ω-special triples arises because not allisogeny classes of abelian varieties contribute to the “twisted” Lefschetz num-ber. The volume terms c(γ0; γ, δ) are the same as those in Kottwitz’ formula.

The term ωp(p−1) is a root of unity in Q

×` depending on ω. The ω-weighted

orbital integral Oωγ (fp) (resp. twisted orbital integral TOω

δ (φr)) differs from theorbital integral Oγ(fp) (resp. the twisted orbital integral TOδ(φr)) appearingin Kottwitz’ formula by a weighting factor coming from ω in the integrand (seeDefinition 6.1 (resp. Definition 6.3)). The term 〈α1(γ0; γ, δ) , κ0〉 is root of

unity in Q×` which results from pairing an element κ0 (which depends on ω)

with a refinement α1(γ0; γ, δ) of the Kottwitz invariant α(γ0; γ, δ). This refinedinvariant is defined for any triple (γ0; γ, δ) for which α(γ0; γ, δ) can be defined.Moreover, it lies in a certain finite abelian group X∗(Z(I1)

Γ)/im(Z×p ) which

maps to the group k(I0/Q)D to which α(γ0; γ, δ) belongs. Moreover, under themap

X∗(Z(I1)Γ)/im(Z×

p ) → k(I0/Q)D,

α1(γ0; γ, δ) maps to α(γ0; γ, δ). It follows from this comparison that if we takeω to be the trivial character in the formula for the twisted Lefschetz number,then we recover the formula of Kottwitz for the untwisted Lefschetz number.

The refined invariant α1(γ0; γ, δ) is constructed so that the roots of unity〈α1(γ0; γ, δ) , κ0〉 keep track of how each isogeny class of abelian varieties con-tributes to the twisted Lefschetz number. If (γ0; γ, δ) comes from the isogenyclass of a polarized abelian variety (A, λ) over the finite field Fpr , then the Kot-twitz invariant α(γ0, γ, δ) is trivial. However, the refined invariant α1(γ0; γ, δ)also reflects the choices of symplectic similitudes between the A

pf -Tate module

(resp. the covariant Dieudonne module) and the standard symplectic space:

βp : H1(A,Apf )

v−→ V ⊗ Apf

βr : H(A)v−→ V ⊗ Lr.

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The refined invariant α1(γ0; γ, δ) of a triple coming from (A, λ) is trivial if andonly if βp and βr are symplectic isomorphisms, not just symplectic similitudes(relative to a fixed symplectic structure on V of course).

The definition and study of α1(γ0; γ, δ) is the technical heart of this pa-per. Both the definition and the proofs of the important properties rest ona theorem of Reimann and Zink [27]. We put their theorem into a group-theoretic framework and thereby construct a refinement of the canonical mapB(GQp

) → X∗(Z(G)Γ(p)) used in the construction of Kottwitz’ invariantα(γ0; γ, δ); this is used to define α1(γ0; γ, δ). We also use the theorem ofReimann and Zink to prove the crucial vanishing property of α1(γ0; γ, δ) al-luded to above, in effect by proving that it implies the vanishing of a “refinedinvariant” attached to polarized abelian varieties over finite fields, and then byrelating this vanishing to that of α1(γ0; γ, δ) when the maps βp (resp βr) aboveare symplectic isomorphisms. Theorem 0.1 then follows from a study of howthe quantity 〈α1(γ0; γ, δ) , κ0〉 changes when the condition that βp (resp. βr)preserves symplectic pairings is relaxed.

The form of the expression for the twisted Lefschetz number of Theorem 0.1is reminiscent of part of the geometric side of the twisted trace formula. In factthe strategy of this paper is modeled on a very special case of the stabilizationof the regular elliptic part of the twisted trace formula, due to Kottwitz andShelstad [13]. This special case results in a formula analogous to Theorem 8.2 ofthis paper, except that our formula includes as well the terms which correspondto the singular elliptic contribution to the twisted trace formula. In particularone sees that the invariant α1(γ0; γ, δ) is the analogue of the invariant obs(δ)from [13]. The reader familiar with [13] will anticipate the next step in theprocess: one should be able to “stabilize” the formula above, that is, write itas a linear combination of the (elliptic parts of the) stable trace formulas forcertain functions fH , where H ranges over all endoscopic groups for the pair(G,a). This should then yield a description of the cohomology of each connectedcomponent of SK , in a manner analogous to §10 of [9]. We will attend to thisaspect of the problem in a future paper.

The results in this paper are related to results of M. Pfau [24], who studieda refinement of the conjecture of Langlands and Rapoport [18] for connectedShimura varieties. They are also related to the book of H. Reimann [26]; inparticular, the “mysterious invariant” in Appendix A of [26] should be relatedto the refined invariant α1(γ0; γ, δ) of this paper. At the present time, however,we are unable to say exactly what the relationships are.

The paper is organized as follows: in §1 we define notation. In §2 we definethe operators L(ω), and reduce the computation of the virtual trace to thecomputation of a sum over fixed points of certain roots of unity. In §3 weprove a formula for the sum over the fixed points in a single isogeny class. Asa necessary prelude to the definition of the refined invariant in §5, in §4 weintroduce a p-adic construction which gives a group-theoretic way to view thetheorem of Reimann and Zink, which is also recalled. In §6 we perform furthersimplifications of the sum in question (essentially determining which isogenyclasses contribute, and then grouping them together into “stable” classes). In

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§8 the Main Theorem is proved, using a crucial vanishing property of the refinedinvariant α1(γ0; γ, δ). This follows from the vanishing of an invariant α(A, λ)attached to polarized abelian varieties, which is deduced from the theorem ofReimann-Zink in §7.

1 Notation and terminology

Let p > 2 be a prime number. Write k for the finite field Fp. Fix an algebraicclosure k of k. For each positive integer r let kr denote the field with pr elementsin k. We will use v to denote a place of Q. Let ` denote a fixed finite place ofQ other than p. We will often use the symbol l to denote an unspecified finiteplace that is not p (the symbols ` and l are always used in different contexts,so this should not cause confusion). Fix an algebraic closure Qp of Qp, and

let Qunp denote the maximal unramified extension of Qp in Qp. Let Q∞ = C.

Write W (k) for the ring of Witt vectors of k. It is the completion of the ringZun

p of integers in Qunp . Let L denote the completion of Qun

p , and write σ

for the Frobenius automorphism of L. Fix an algebraic closure L of L whichcontains Qp. For each r ≥ 1 let Lr denote the fixed field of σr in L. Then Lr

is the unique unramified extension of Qp in L having degree r, and this is alsosometimes denoted Qpr .

If F ⊂ R, we will use F+ to denote F ∩ R>0.The crucial constructions in this paper can only be done modulo the choice

of the following initial data D = i , jp, j∞. Here i =√−1 is a choice of

square root of −1 in C, and jp : Q → Qp and j∞ : Q → C are choices of fieldembeddings.

By using the data D and the “reduction modulo p” map Zunp → k we get,

for each l 6= p and n ∈ N, identifications of roots of unity:

µln(k) = µln(Zunp ) = µln(Qp) = µln(Q) = µln(C) = Z/(lnZ).

Here the last identification is defined by the inverse of the exponential map:x 7→ exp((2πix)/ln), where i is the element fixed in D. By taking the limit overn and then tensoring with Q we get identifications of Tate twists

Ql(1)(k) = Ql(1)(Qp) = Ql(1)(Q) = Ql(1)(C) = Ql,

and so after taking the restricted product over all l 6= p, we get identifications

Apf (1)(k) = A

pf (1)(Qp) = A

pf (1)(Q) = A

pf (1)(C) = A

pf .

In a similar way, the data D give us identifications for the prime p:

Qp(1)(Qp) = Qp(1)(Q) = Qp(1)(C) = Qp.

Write Γ for Gal(Q/Q). If v is any place of Q and we have chosen an algebraicclosure Qv of Qv, then we write Γ(v) for Gal(Qv/Qv). Any choice of field

6

embedding jv : Q → Qv yields an inclusion j∗v : Γ(v) → Γ; whenever we areconsidering choices of such jv for v ∈ p,∞, we will use the choices alreadyspecified in the data D. Throughout the paper we will use the symbol j∗v quitegenerally to denote any map between objects that is defined in a transparentmanner by using the embedding jv.

Now we describe the Shimura varieties SKp in this paper, following theintroduction of [11]. Let V be a 2g-dimensional nondegenerate symplectic vectorspace over Q (a finite dimensional Q-vector space together with a nondegenerateQ-valued alternating pairing 〈·, ·〉). Let G be the group of automorphisms ofthe symplectic space V . Since automorphisms are only required to preserve thepairing up to a scalar, G is a realization of the group of symplectic similitudesGSp(2g,Q). Let Kp ⊂ G(Ap

f ) be a compact open subgroup which is sufficientlysmall (the precise meaning of which we describe below). Let h : C → End(VR) bean R-algebra homomorphism such that h(z) = (h(z))∗ and 〈·, h(i)·〉 is positivedefinite, where ∗ denotes the transpose on End(VR) coming from the pairing,and i is from the data D. Let X∞ denote the G(R)-conjugacy class of h. LetK∞ denote the subgroup of G(R) centralizing h. Let X+ denote the set of allelements h1 ∈ X∞ such that 〈·, h1(i)·〉 is positive definite (again use the i in D).

Now consider the following moduli problem SKp over Z(p): for a locallyNoetherian Z(p)-scheme S let SKp(S) denote the set of all isomorphism classes

of triples (A, λ, η), where A is a projective abelian scheme over S, λ : A→ A is apolarization of A (a prime-to-p isogeny which is a polarization over all geometricpoints of S), and η is a Kp-orbit of symplectic similitudes

η : V ⊗ Apf → H1(A,A

pf ),

where the pairing on the right is the Weil pairing coming from the polarizationλ. Here we are viewing both sides as smooth A

pf -sheaves over S with the etale

topology, and we are assuming that for each geometric point s of S, the Kp-orbitηs of the map on stalks ηs is fixed by the action (on the left) of the algebraicfundamental group π1(S, s). Such an orbit η will be called a Kp-level structure.We say two triples (A, λ, η) and (A′, λ′, η′) are isomorphic if there exists a prime-to-p isogeny φ : A → A′ such that φ∗(λ′) = cλ, for some c ∈ Z+

(p), and which

takes η to η′.In [23] Mumford used geometric invariant theory to prove that SKp is rep-

resented by a smooth quasiprojective Z(p)-scheme, for any Kp which is smallenough so that the moduli problem SKp has no nontrivial automorphisms (thisis what we mean when we say Kp is “sufficiently small”). Kottwitz generalizedthis result in §5 of [11].

We fix additional data: let Λ0 ⊂ V be a Z(p)-lattice which is self-dual withrespect to the pairing, and let Kp denote the stabilizer of this lattice in G(Qp).Then Kp is a hyperspecial good maximal compact subgroup of G(Qp) and wecan choose an integral structure for GQp

such thatKp = G(Zp). LetK = KpKp.We will often write SK instead of SKp .

Write Gsc for the derived group of G, which is simply connected since it isa realization of Sp(2g,Q). Let c : G→ Gm denote the homomorphism given by

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the equation 〈gv, gw〉 = c(g)〈v, w〉.We denote by π0 the scheme of connected components of SK . It is a finite

etale Z(p)-scheme.

If G is any connected reductive group over a field F , we denote by G theconnected reductive group over C of type dual to G, together with the action ofGal(F sep/F ) on G (which is well-defined up to an inner automorphism of G).Denote by Z(G) the center of G; it carries a canonical action of Gal(F sep/F ).

2 The twisted Lefschetz number

2.1 Definition of the Operator L(ω)

Let πp denote the finite abelian group

Z+(p)\(A

pf )×/c(Kp) ∼= Q+\A×

f /c(Kp)Z×

p∼= Q×\A×/c(K)c(K∞).

(One can show without difficulty that c(Kp) = Z×p and that c(K∞) = R+ .)

Fix a character ω : πp → C×. Sometimes (but not always) we will need toconsider this character as Q`-valued. To do so we choose an isomorphism offields C

v−→ Q`; the context will dictate in which field we are considering ω totake values.

Next we will construct maps (depending on D) c : SK(F ) → πp, where F isany of the rings k, Zun

p , Qp, Q, or C, and then we will denote by ω any of theresulting maps ω c : SK(F ) → C×. Because π0 is finite and etale, D yields thefollowing commutative diagram

SK(k)

SK(Zunp )oo

// SK(Qp)

SK(Q)oo

// SK(C)

v // G(Q)\G(A)/KK∞

c

π0(k) π0(Zunp ) π0(Qp) π0(Q) π0(C)

v // πp.

The rightmost square results from the fact that if we specify h ∈ X+ as a basepoint, then we can identify the map

c : G(Q)\G(A)/KK∞ −→ πp

induced by the homomorphism c : G(A) → A× with the canonical map

SK(C) −→ π0(C),

see [1]. Using this diagram we can define c : SK(F ) → πp for any of the ringsF .

Definition 2.1. Let F be any of the rings k, Zunp , Qp, Q, or C. For each

character ω : πp → Q×` , define the operator L(ω) on H i

c(SK⊗F , Q`) to be⊕

a∈πp

ω(a) :⊕

a∈πp

Hic(c

−1(a) , Q`) −→⊕

a∈πp

Hic(c

−1(a) , Q`).

8

Next we discuss Hecke correspondences. Let g ∈ G(Apf ), and define Kp

g =

Kp ∩ gKpg−1. We denote by f the Hecke correspondence on the scheme SKp

SKp SKpg

R(g)oo

R(1)// SKp

where R(1) is the canonical projection coming from the inclusion Kpg ⊂ Kp, and

R(g) comes from right translation by g; on points R(g) is given by [A, λ, η] 7→[A, λ, ηg].

Let F be one of the rings k, Zunp , Qp, Q, or C. If we view Q` as the trivial

local system on SK ⊗F , there is a canonical morphism R(g)∗(Q`) → R(1)!(Q`)of `-adic local systems on SKp

g⊗ F . Then the correspondence f together with

this choice of morphism R(g)∗(Q`) → R(1)!(Q`) determines an operator f can

on cohomology groups with compact support

f = f can : Hic(SK⊗F , Q`) → Hi

c(SK⊗F , Q`).

We can also replace the canonical morphism can : R(g)∗(Q`) → R(1)!(Q`) withthe composition

R(g)∗(Q`)ωc // R(g)∗(Q`)

can // R(1)!(Q`) ,

where the second map is the canonical one and the first is the map of sheavesover SKp

gwhich is given on the stalk over x ∈ SKp

g(F ) by multiplication by

ω c(R(g)(x)), where c : SK(F ) → πp is as above. Thus f together with ωdetermines a map on cohomology with compact supports which we denote byfω.

Lemma 2.2. As operators on cohomology groups with compact supportsHi

c(SK⊗F , Q`), we havefω = f L(ω),

where F is k, Zunp , Qp, Q, or C.

Proof. This follows directly from the definition of how (finite, flat) correspon-dences induce maps on cohomology with compact supports. See [2], p. 210 forthe definition.

Now let σp denote a choice of arithmetic Frobenius in Gal(Qp/Qp) and sup-

pose F is Zunp , Qp, or Q. By transport of structure σ−1

p defines an operator

on Hic(SK ⊗ F , Q`). Since SK has good reduction at p, the inertia subgroup

at p acts trivially on the cohomology groups H ic(SK ⊗ F , Q`), and so the op-

erator is independent of the choice of σp. Thus we have a well-defined operatorσ−1

p fω = σ−1p f L(ω) acting on H i

c(SK⊗F , Q`).

Now suppose F = k. Let Φp be the Frobenius morphism for the k-schemeSK . We get the correspondence fr := Φr

p f by replacing the morphism R(1) inthe definition of f with R(1)r := Φr

p R(1). If we use the canonical morphism

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can : R(g)∗(Q`) → R(1)!r(Q`) (viewing Q` as the trivial local system on SK),then we get an operator denoted by the same symbol

fr = Φrp f : Hi

c(SK⊗F , Q`) −→ Hic(SK⊗F , Q`),

(the r-th “twist” of f). We can also replace the morphism can : R(g)∗(Q`) →R(1)!r(Q`) with the composition

R(g)∗(Q`)ωc // R(g)∗(Q`)

can // R(1)!r(Q`) ,

where the second map is the canonical one and the first is the map of sheavesover SKp

gwhich is defined above. As in the “untwisted” case the pair (fr, ω)

determines an operator on cohomology with compact supports which we denoteby (fr)

ω = (Φrp f)ω. Performing the “twists” by ω and Φr

p in the other orderyields an operator (fω)r = Φr

p fω, and it is easy to see that (fr)ω = (fω)r.

To summarize we have the following consequence of our definitions and thereasoning in the proof of Lemma 2.2.

Lemma 2.3. As operators on cohomology groups with compact supportsHi

c(SK⊗k , Q`), we have

Φrp fω = (Φr

p f)ω = Φrp f L(ω).

Remark 2.4. In the case where F = Zunp , Qp, or Q, we can view σ−1

p as amorphism SK ⊗F → SK ⊗F (determined by σ−1

p : F → F via base-change).Thus it makes sense to think of the operator on cohomology σ−1

p f as comingfrom a correspondence, in a way analogous to the case over the finite field.From now on we will abuse notation and denote this correspondence and itsaction on H i

c(SK⊗F , Q`) by the symbol Φrp f . By reasoning analogous to that

preceding Lemma 2.3, we therefore have the formulas σ−rp f = Φr

p f andσ−r

p fω = Φrp fω = (Φr

p f)ω. With our notational conventions the above

lemma holds also when k is replaced by F = Zunp , Qp, or Q.

We have taken care in the above discussion in order to insure that the basechange theorems for etale cohomology with compact supports of SK can beapplied to the situation where the operator L(ω) is present. These are usuallystated for a correspondence such as Φr

p f above, the morphism R(g)∗(Q`) →R(1)!r(Q`) being tacitly assumed to be the canonical one. However, everythingworks just as well when the morphism determined by ω is used instead, i.e, forthe “correspondence” (Φr

p f)ω (see the Appendix of [21]). Thus after inter-preting our original operator σ−r

p f L(ω) as a kind of correspondence we areable to pass to the special fiber and “count fixed points” of the correspondenceover the finite field, in the usual way. This is the content of the next section.

2.2 A first reduction

The base change theorems of etale cohomology with compact supports (see [20],VI, Thm. 3.2) applied to the correspondence (Φr

p f)ω imply that

Tr(Φrp f L(ω) ; H•

c (SK⊗Q , Q`)) = Tr(Φrp f L(ω) ; H•

c (SK⊗k , Q`)).

10

Recall that on the right hand side Φp denotes a geometric Frobenius. To evaluatethe right hand side for sufficiently large r, we will use Deligne’s conjecture, whichhas been proved by Fujiwara [3]. It follows from the statement of Deligne’sconjecture on p.212 of [2] that the right hand side of the above equality is given,for sufficiently high values of r (depending on g), by the sum

∑

[A,λ,η]

ω[A, λ, ηg].

Here we sum over fixed points of Φrpf in SKp

g(k) the value of [A, λ, ηg] ∈ SKp(k)

under the map ω. It is easy to see that this can be rewritten as

ω(g)∑

[A,λ,η]

ω[A, λ, η],

where here we sum over the same set of fixed points the value of [A, λ, η] ∈SKp

g(k) under ω.

Lemma 2.5. Without loss of generality, we can assume ω(g) = 1.

Proof. It is easy to prove by direct calculation on the level of de Rham coho-mology classes that f L(ω) = ω(g)L(ω) f . The same relation holds on etalecohomology groups. Therefore the virtual trace of Φr

p f L(ω) is zero unlessω(g) = 1.

Remark. Looking ahead to the final expression for this sum (Theorem 8.2) wecan easily see that if ω(g) 6= 1, then Oω

γ (fp) = 0, for each orbital integral thatappears in the sum. To prove this, note that f p = char(Kpg−1Kp) and assumethat this orbital integral is nonzero. Then there exists y ∈ G(Ap

f ) such that

y−1γy ∈ Kpg−1Kp. Since ω(γ) = 1 (as ω is trivial on G(Apf )γ), and ω(Kp) = 1,

we see immediately that ω(g) = 1.

We have now achieved the first reduction of our problem: in order to computethe virtual trace of Φr

p f L(ω) on H•c (SK ⊗Q , Q`) (for sufficiently high r),

we need to find a formula for the sum∑

[A,λ,η]

ω[A, λ, η],

where we are summing over the fixed point set Fix in SKpg(k) of the correspon-

dence Φrp f . We may group these fixed points together according to Q-isogeny

class, and then the first step is to find a formula for the portion of the sumcorresponding to a single isogeny class. This is done in the next section.

3 The Fixed Points in a Given Isogeny Class

We need to recall some definitions from §10 of [11], in particular the notionof a c-polarized virtual abelian variety (A, λ) over kr up to prime-to-p isogeny.

11

A virtual abelian variety A over kr up to prime-to-p isogeny consists of a pairA = (A, u) where A is an abelian variety over k up to prime-to-p isogeny andu : σr(A) → A is a prime-to-p isogeny (here σ : k → k is x 7→ xp and σr(A)is defined by base-changing A via σr). We define the Frobenius element πA ∈End(A) by πA = u Φr, where Φr : A → σr(A) is the relative Frobeniusmorphism. Thus if A is defined over kr then the canonical u is an isomorphismand πA is the absolute Frobenius morphism coming from the kr-structure of A.

Let c be a rational number which we write as c = c0pr. By a c-polarization

of A we mean a Q-polarization λ ∈ Hom(A, A) ⊗ Q such that π∗A(λ) = cλ.

Equivalently, u∗(λ) = c0σr(λ). In order for λ to exist it is necessary that

c ∈ Q+ and that c0 is a p-adic unit.As in §10 of [11], attached to (A, λ) we have the A

pf -Tate module H1(A,A

pf ),

on which πA acts. Also, we have an Lr-isocrystal (H(A),Φ) which is defined tobe the u-fixed points in the L-isocrystal (H(A),Φ) dual to H1

crys(A/W (k))⊗Ltogether with its σ-linear bijection. The morphism πA also induces endomor-phisms of H(A) and H(A). On H(A) we have the identification Φr = π−1

A u,hence on H(A) we have Φr = π−1

A .

Each fixed point [A′, λ′, η′] ∈ SKp

g(k) of the correspondence Φr

p f gives riseto a c-polarized virtual abelian variety over kr up to prime-to-p isogeny (A′, λ′)(see §16 of [11]). Let us fix one (A, λ) coming from a fixed point and let S denotethe set of those fixed points [A′, λ′, η′] ∈ SKp

g(k) which have the property that

(A′, λ′) and (A, λ) are Q-isogenous. The purpose of this section is to prove aformula for ∑

[A′,λ′,η′]∈S

ω[A′, λ′, η′].

Recall that in the statement of the moduli problem for SKp we are given Vwith a symplectic pairing 〈·, ·〉 and a self-dual Zp-lattice Λ0 ⊆ V ⊗Qp. Now wechoose symplectic similitudes

βp : H1(A,Apf )

v−→ V ⊗ Apf

βr : H(A)v−→ V ⊗ Lr.

Here is what we mean by the term “symplectic similitudes”: H1(A,Apf ) is the

Apf -Tate module for the k-abelian variety A. We can regard the pairing (·, ·)λ on

this space induced by the polarization λ as having values in Apf as follows: the

pairing originally takes values in Apf (1)(k). Using the data D we can identify

this with Apf as in §1.4, and thus we can consider the pairings on both sides

as Apf -valued. We call βp a symplectic similitude if it preserves the pairings up

to a scalar: (βp)∗〈·, ·〉 = c(βp)(·, ·)λ, for some c(βp) ∈ (Apf )×. The property of

being a symplectic similitude is independent of the choice of data D, althoughof course the value of c(βp) is not. Furthermore, the polarization λ induces anL-valued pairing (·, ·)λ on the L-isocrystal (H(A),Φ) (see §10 of [11]). Once wechoose d ∈ O×

L such that d−1σr(d) = c0, then the pairing d(·, ·)λ is Lr-valuedon the Lr-isocrystal (H(A),Φ), and this pairing is well-defined up to an element

12

of O×Lr

. We call βr a symplectic similitude if there exists c(βr) ∈ L×r such that

β∗r 〈·, ·〉 = c(βr)d(·, ·)λ.

Again the propery of being a symplectic similitude is independent of the choiceof d, but the value of c(βr) ∈ L×

r is not. However, the image of c(βr) inL×

r /O×Lr

is independent of the choice of d. We will often simply write β for thepair (βp, βr).

Let I = Aut(A, λ), and define Xp to be the set of Kpg -orbits of symplectic

similitudesη : V ⊗ A

pf −→ H1(A,A

pf )

such that πAη ≡ ηg (mod Kp). Similarly we define Xp to be the set of latticesΛ ⊂ H(A) which are self-dual up to a scalar in L×

r , and for which we have

p−1Λ ⊃ ΦΛ ⊃ Λ.

We will consider the following diagram:

S

c

∼ // I(Q)\[Xp ×Xp]

cp×cp

Z+(p)\(A

pf )×/c(Kp) ∼ // Q+\A×

f /c(Kp)Z×

p

We will need to discuss each of the four maps in turn.1. The top map may be viewed as the first step in the usual process of writingthe fixed points of a correspondence in a single isogeny class in terms of cosetsfor the groups G(Ap

f ) and G(Lr). We describe it explicitly in the following way.First choose a Q-isogeny φ : (A′, λ′) → (A, λ). Then we define the map to be

[A′, λ′, η′] 7→ [H1(φ) η′, H(φ)D(A′)].

Here D(A′) ⊆ H(A′) is the usual lattice inside the Lr-vector space H(A′) (see§10 of [11]). This map is clearly independent of the choice of φ.2. The left map is given as follows: We define c([A′, λ′, η′]) by considering theelement c of Z+

(p)\(Apf )×/c(Kp

g ) such that

η′∗(·, ·)λ′ = c〈·, ·〉.

Here (·, ·)λ′ is the Apf -valued pairing (using D to consider it as such) onH1(A′,Ap

f )

coming from λ′, and 〈·, ·〉 is the given pairing on V . Take c([A′, λ′, η′]) to be theimage of c in Z+

(p)\(Apf )×/c(Kp).

3. The right map: Consider a Kpg -orbit of symplectic similitudes

η : V ⊗ Apf −→ H1(A,A

pf ),

and a lattice Λ ⊂ H(A) which is self-dual up to a scalar. Then cpg(η) ∈

(Apf )×/c(Kp

g ) is defined by the equality

η∗(·, ·)λ = cpg(η)〈·, ·〉,

13

and cp(η) is the image of this element in (Apf )×/c(Kp). In a similar manner we

define cp(Λ) ∈ L×r /O×

Lr= Q×

p /Z×p by

Λ⊥ = cp(Λ)−1Λ.

4. The bottom isomorphism is induced by the canonical inclusion of Apf×

into

A×f .

One can check easily that this diagram commutes. Note also that the entirediagram is independent of the choice of β.

Remark 3.1. The commutativity of this diagram is related to the commu-tativity of the diagram at the beginning of §2 which was used to define themap c : SK(k) → πp. In fact under the indentification of π0(SK)(k) ∼= πp,the left vertical map c : S → πp above is simply the restriction to S of themap SKp

g(k) → π0(SKp

g)(k) → π0(SKp)(k). This provides an interpretation of

c : SK → π0(SK) “on points” and explains why we used the same symbol c forthe vertical map in the above diagram.

Next we will consider a second diagram wherein the effect of altering βcomes clearly into view. Using βp we can transport the action (by symplecticsimilitudes) of π−1

A on H1(A,Apf ) over to V ⊗A

pf . Denote the resulting element

of G(Apf ) by γ. Similarly use βr and the σ-linear bijection Φ : H(A) → H(A) to

define δσ, where δ ∈ G(Lr). Note that γ and δ depend on our choice of β, butaltering β does not change the G(Ap

f )-conjugacy class of γ or the σ-conjugacyclass of δ in G(Lr).Define the sets:

Y p =y ∈ G(Ap

f )/Kpg

∣∣ y−1γy ∈ Kpg−1,

Yp =x ∈ G(Lr)/Kr

∣∣ x−1δσ(x) ∈ Krσ(a)Kr

.

Here Kr is the stabilizer in G(Lr) of the lattice Λ0⊗Lr, and a = µ1(p)−1, where

µ1 comes from h ∈ X∞ as on p.430 of [11]. The double coset Krσ(a)Kr doesnot depend on the choice of h. We note that the sets Y p and Yp both dependon the choice of β, since γ and δ do. We need to explain the following diagram:

I(Q)\[Xp ×Xp]∼ //

cp×cp

βI(Q)β−1\[Y p × Yp]

c

Q+\A×f /c(K

p)Z×p

∼c(βp)c(βr)

// Q+\A×f /c(K

p)Z×p

1. The top isomorphism is [η,Λ] 7→ [yKpg , xKr]. Here y is any representative for

the Kpg -orbit of symplectic similitudes

βpη : V ⊗ Apf −→ V ⊗ A

pf .

14

Note that βr(Λ) is a lattice in V ⊗ Lr which is self-dual up to a scalar. ByCorollary 7.3 of [11], there is a uniquely determined x ∈ G(Lr)/Kr such thatβr(Λ) = x(Λ0). This gives us the coset xKr.

2. The right map is [yKpg , xKr] 7→ [c(y)c(x)]. Here c(y) ∈ A

pf×/c(Kp

g ) and

c(x) ∈ Q×p /Z

×p , the image of c(x) ∈ L×

r /O×Lr

under the identification

L×r /O×

Lr= Z = Q×

p /Z×p .

The brackets [. . .] mean take the class of (c(y), c(x)) in Q+\Apf/c(K

p)Z×p .

3. The bottom isomorphism is multiplication by c(βp)c(βr). Here c(βr) denotesthe image of c(βr) ∈ L×

r /O×Lr

under the identification L×r /O×

Lr= Q×

p /Z×p .

One can easily check that this second diagram commutes, keeping in mind theidentities

(xΛ0)⊥ = c(x)−1(xΛ0) (∀x ∈ G(Lr)),

(βrΛ)⊥ = c(βr)−1cp(Λ)−1(βrΛ).

Since xΛ0 = βrΛ (by definition of x), we see that c(x) = c(βr)cp(Λ) in L×r /O×

Lr.

The rest of the verification is easy.Now as in §16 of [11], we let dy denote the Haar measure on G(Ap

f ) givingKp

g measure 1. Let dx denote the Haar measure on G(Lr) giving Kr measure 1.

Let fp denote the characteristic function of Kpg−1, and let φr denote the char-acteristic function of Krσ(a)Kr. We use the Haar measure on the discrete groupβI(Q)β−1 that give points measure 1. Now using the commutativity of the twodiagrams above and standard calculations (keeping in mind that Kp is suffi-ciently small that the moduli problem SKp has no nontrivial automorphisms)we can conclude with the following theorem:

Theorem 3.2. In the notation above the sum

∑

[A′,λ′,η′]∈S

ω[A′, x, η′]

can be written as

ω (c(βp)c(βr))−1∫

βI(Q)β−1\[G(Apf)×G(Lr)]

ω (c(y)c(x)) fp(y−1γy)φr(x−1δσ(x)) dydx.

To avoid overly complicated notation from this point on, we write the integralmore simply as

ω(βpβr)−1

∫

I(Q)\[G(Apf)×G(Lr)]

ω(y)ω(x)fp(y−1γy)φr(x−1δσ(x)) dydx,

and we write the sum above as T ω(A, λ).

15

4 Construction of Some Useful Maps

4.1

For this section we make a slight change in our notation. Let F denote a p-adic field, and L the completion of the maximal unramified extension of F insome algebraic closure F . Let L be an algebraic closure of L which containsF . Let Γ = Gal(F/F ), and denote by σ the Frobenius automorphism of L overF . If G is any connected reductive F -group let B(G) denote the pointed set ofσ-conjugacy classes in G(L), as defined by Kottwitz in [7].

Let T be any F -torus. In §2 of [7] there is the construction of a canonicalfunctorial isomorphism

B(T )v−→ X∗(T )Γ.

This is canonical up to the choice of generator of Z. Note that B(Gm) = Z andthat the isomorphism is determined by functoriality once we set the conventionthat X∗(Gm) → B(Gm) be µ 7→ [µ(p)]. To get the other isomorphism wewould have used the other generator of B(Gm), namely the σ-conjugacy classof elements of L× having normalized valuation −1. This would give a differentfunctorial isomorphism, and the map X∗(Gm) → B(Gm) would then be µ 7→[µ(p−1)].

The aim of this section is to define groups Bπ(T1 → T ) and X(T1 → T )attached to any exact sequence of F -tori

1 //T1//T

c //Gm//1

and to construct an isomorphism Bπ(T1 → T ) −→ X(T1 → T ) that will dependon a tower of uniformizers Π. This construction was inspired by a similar onedue to Reimann and Zink [27], and the point here is to put their constructioninto a group-theoretic framework.

We proceed to the relevant definitions. Fix a uniformizer π of L. Let usconsider the collection of all finite Galois extensions E of L in L. Suppose

Π = πE is a collection of uniformizers πE of E, one for each E.

Definition 4.1. We call Π a tower over π if

1. E′ ⊇ E ⇒ NmE′/E(π′E) = πE ,

2. πL = π.

Lemma 4.2. For each π, there is a tower Π over π.

Proof. If we are given E′ ⊃ E and a uniformizer πE of E, then there exists auniformizer πE′ of E′ such thatNmE′/E(πE′) = πE . To see this use the fact thatNmE′/E(UE′) = UE , a consequence of the fact that E ′/E is totally ramified (see

V §6 of [28]). Therefore we will be done if we can show that L/L is exhausted by acountable family of finite Galois extensions Ei/L. But F , being p-adic, has onlya finite number of Galois extensions of given degree, so F/F is exhausted by acountable collection of finite Galois extensions Fi/F , and thus so is F/F un (use

16

FiFun). But now our lemma follows from the observation that L/L and F/F un

have isomorphic lattices of intermediate fields, since Gal(L/L) = Gal(F/F un).

Consider next an exact sequence of F -tori

1 //T1//T

c //Gm//1 .

This is still exact on L-points, as H1(L, T1) = (1).

Definition 4.3. For each uniformizer π of L, define

1. T (L)π = c−1(πZ),

2. Bπ(T1 → T ) = T (L)π/(A〈σ〉T1(L)).

Here A〈σ〉 denotes the augmentation ideal for the infinite cyclic group 〈σ〉.Therefore we get an exact sequence

1 //T1(L) //T (L)πc //πZ //1 ,

and by dividing out each of the first two groups by A〈σ〉T1(L), we get the exactsequence

1 //B(T1) //Bπ(T1 → T )c //πZ //1 .

Definition 4.4.

X(T1 → T ) =X∗(T )

A(Γ)X∗(T1).

Here A(Γ) denotes the augmentation ideal for the group Γ. Now an argumentsimilar to the one above shows that there is an exact sequence

0 //X∗(T1)Γ //X(T1 → T )c //X∗(Gm) //0 .

Claim 4.5. For any tower Π over π and exact sequence of F -tori as above, wecan define a group homomorphism

KΠ : X∗(T ) −→ Bπ(T1 → T )

as follows: Choose any E/L that splits T (and thus T1) and let KΠ send µ ∈X∗(T ) to the class of NmE/L(µ(πE)) in Bπ(T1 → T ). It is easy to see that thisis well-defined using the properties (1) and (2) in Definition 4.1.

Proposition 4.6. KΠ has the following properties:

1. KΠ factors through X(T1 → T ).

17

2. The following diagram commutes:

0 // X∗(T1)Γ

// X(T1 → T )

KΠ

c // X∗(Gm)

// 0

1 // B(T1) // Bπ(T1 → T )c // πZ // 1

.

where the left vertical arrow is Kottwitz’ map in [7], and the right verticalarrow is n 7→ πn.

Proof. Restricting KΠ to X∗(T1) gives exactly Kottwitz’ map X∗(T1) → B(T1),as is seen easily from the definition. Since Kottwitz’ map is trivial onA(Γ)X∗(T1),so is KΠ. This proves (1) and also the commutativity of the left square of (2).The commutativity of the right follows from the functoriality (with respect toT ) of the map X∗(T ) → T (L) given by Π.

Corollary 4.7.

KΠ : X(T1 → T ) → Bπ(T1 → T )

is an isomorphism.

Proof. The other two vertical arrows above are isomorphisms.

Now we want to construct a generalization of K−1Π that works for connected

reductive groups instead of just tori. Consider an exact sequence of connectedreductive F -groups

1 //I1 //Ic //Gm

//1.

Definition 4.8.

I(L)π = c−1(πZ).

Definition 4.9.

Bπ(I1 → I) =I(L)π

A〈σ〉I1(L).

Definition 4.10.

X(I1 → I) =X∗(Z(I))

A(Γ)X∗(Z(I1)).

In the second definition we are abusing notation, and the quotient is meant todenote the equivalence relation

x ∼ y ⇐⇒ x = z−1yσ(z)

for some z ∈ I1(L). Thus Bπ(I1 → I) is only a pointed set and not a group ingeneral. In the third definition I denotes the connected dual group (over C). IfI is a torus this definition agrees with the earlier one.

18

We want to construct a functorial surjection of pointed sets

MΠ : Bπ(I1 → I) −→ X(I1 → I)

which agrees withK−1Π when I is a torus. To do so, first assume that Ider = Isc.

Let D = I/Ider, and D1 = I1/Ider. Note that D and D1 are both tori. Also, c

induces a map c : D → Gm. We have the following commutative diagram withexact rows and columns:

1 // I1 //

Ic //

Gm

id

// 1

1 // D1// D

c // Gm// 1

On L-points the sequences are still exact, and I(L)π → D(L)π is a surjection.

Since Ider is simply connected, we know that the restriction map X∗(Z(I)) →X∗(D) is an isomorphism, and it follows from this that the same is true withI1 replacing I and D1 replacing D. Thus X(I1 → I) = X(D1 → D) and wecan define

MΠ : I(L)π −→ X(I1 → I)

to be the unique map making the following diagram commute

I(L)π

// X(I1 → I)

D(L)π

K−1

Π // X(D1 → D).

The mapMΠ is obviously functorial in exact sequences like the one containing I1and I, and it is equally clear that it is a surjective group homomorphism whichagrees with K−1

Π when I is a torus. By properties of K−1Π already mentioned

we can deduce that MΠ factors through Bπ(I1 → I).

Remark. In the applications of this construction we will be concerned withthe case where I is the centralizer of a semisimple element of G(Q) and whereI1 = I ∩Gsc. Since G = GSp(2g) we know not only that I and I1 are connected

reductive groups but also that Ider = Ider1 is simply connected, so that the

above construction will suffice for our purposes. However, the construction canbe made to work for I1 and I arbitrary connected reductive groups by passingto a z-extension I ′ of I: we end up replacing the sequence with I and I1 with

one involving I ′ and I ′1 = ker(I ′ //Ic //Gm ), and then using the usual

argument.

4.2 The Theorem of Reimann and Zink

In this section we will recall the main result of [27]. We will first change ournotation once again. We will now denote by F a CM-algebra with involution

19

∗. By this we mean that F is a product of totally imaginary extensions oftotally real number fields, and that ∗ induces complex conjugation on eachfactor of F ⊗ R ∼= C × · · · × C. Let L denote the completion of a maximalunramified extension of Qp as in §1. Recall that we have fixed i =

√−1 ∈ C

and embeddings jp, j∞. Given these data, Reimann and Zink define an elementg ∈ L(

√p) as follows. Let γ ∈ C be the unique solution in R+ q iR+ for the

equation γ2 = (−1)(p−1)/2p. Using jp and j∞ we regard γ as an element of L.Let [(

p− 1

2

)!

]∈ W (k)

be the unique root of unity which is congruent modulo p to ((p− 1)/2)!. We set

g = −[(

p− 1

2

)!

]−1

γ.

Note that g2 = −p, so that NmL(√

p)/L(g) = p.

Now to our CM-algebra F we associate the Q-torus T = x ∈ F×|xx∗ ∈ Q×.Let T1 be the Q-torus x ∈ F×|xx∗ = 1. Let

S ⊂ HomQ(F,Qp)

be a CM-type for F , with associated cocharacter µS ∈ X∗(TQp). Using jp, j∞

we can also consider this to be in X∗(TC).

Lemma 4.11. For any such F, T, µS as above,

KΠ(µS) ∈ Bp(T1 → T )

is independent of the choice of tower Π over g.

Proof. This is proved, line by line, by following the argument of Reimann andZink on p.472 of [27].

Remark. In fact an inspection of the argument cited above shows that we mayreplace g with any uniformizer in L(

√p). This yields the following:

Corollary 4.12. Fix πL(√

p) a uniformizer of L(√p) with Nm(πL(

√p)) = π,

where π is a uniformizer of L. Let F and T be as above. Then

KΠ(µ) ∈ Bπ(T1 → T )

is independent of the choice of tower Π over πL(√

p), for all µ ∈ X∗(TQp).

Proof. Consider the commutative diagram with exact rows

0 // X∗(T1)Γ(p)

// X(T1 → T )

KΠ

c // X∗(Gm)

// 0

0 // B(T1) // Bπ(T1 → T )c // (π)Z // 1

20

Note that the unlabelled vertical arrows are both isomorphisms and are indepen-dent of the choice of Π over πL(

√p). Also, both these exact sequences split, and

there are particular elements that give the splitting, namely µS and KΠ(µS),since c[µS ] = id, and c[KΠ(µS)] = π. Because these elements are independentof Π (over πL(

√p)), the splittings are independent of the choice Π, and every

KΠ preserves the splittings. Thus KΠ, as a map, is independent of the choiceof Π over πL(

√p).

Remark. ¿From now on, when considering tori T as above and some uni-formizer g′ = πL(

√p) ∈ L(

√p), we will use the notation Kg′ in place of KΠ,

where Π is a tower over g′. The corollary above justifies this notation.

We now recall the statement of the main theorem of [27]. Suppose Ais an abelian variety over the integers of a p-adic field K, and let Tp(A) =H1(AQp

,Zp), the p-adic Tate module of AQp. Let A denote the geometric spe-

cial fiber Ak. We can associate to A an L-isocrystal (H(A),Φ); by definition this

is the dual of the L-isocrystal (H1crys(A/W (k))⊗L,Fr), where Fr denotes the

Frobenius map on crystalline cohomology. Suppose that A has complex multi-plication by the CM-algebra F , and that µCM is the associated cocharacter ofthe Q-torus T = x ∈ F×|xx∗ ∈ Q×. Suppose also that A has a polarizationλ : A → A which commutes with the F -action; this gives pairings (·, ·)λ onH(A) and Tp(A) ⊗ L. The pairing on Tp(A) ⊗ Q is a priori Qp(1)(Qp)-valued,but it can viewed as Qp-valued using our data D and exp(2πix) for our fixedchoice of i ∈ C, as explained in §1. Choose a symplectic isomorphism over L

H(A)v−→ Tp(A) ⊗ L

and use it to define b0 ∈ Bp(T1 → T ) by the equality

Φ−1 = σ−1b0.

Theorem 4.13. (Reimann-Zink) With the particular element g and the othernotation as above, Kg[µCM ] = [b0].

Proof. See Satz 1.8 of [27].

We remark that the hypothesis p > 2 is necessary in the results of this paperonly because the same restriction occurs in the theorem of Reimann and Zinkabove.

4.3

Suppose I is the centralizer of a semisimple element of G(Q) and that T is amaximal Q-torus in I. We showed in the previous section (§4.2) that if thetorus is the group of automorphisms of a CM-algebra with involution ∗, thenthe maps KΠ for T depend only on the choice of πL(

√p). We will need to know

that the same is true for the maps MΠ defined for the groups I. First we remarkthat I always contains such a torus T ; the reader not familiar with this fact can

21

look at the proof of the Main Theorem in §8, where such a torus inside I isconstructed. Next consider the diagram

T (L)π

K−1

Π

// I(L)π

MΠ

X(T1 → T ) // X(I1 → I)

Proposition 4.14. MΠ is uniquely determined on such I by πL(√

p).

Proof. Let x ∈ I(L)π. Choose t ∈ T (L)π having the same image in πZ as x.Thus t−1x ∈ I1(L). Note that restricting MΠ to I1(L) gives the canonical map

wI1: I1(L) −→ X∗(Z(I1))Γ

constructed in §7 of [12]. (To check this we use that I1/Ider1 → D1, in the

notation used earlier.) Now our result follows from the equation

MΠ(x) = K−1Π (t) + wI1

(t−1x).

Remark. From now on, when we consider such a centralizer I, a uniformizerg′ ∈ L(

√p), and a tower Π over g′, we will write Mg′ instead of MΠ. The

proposition above justifies this notation.

We will use the maps Mg, for the particular element g of Reimann-Zink con-structed above, to define the invariants α1(γ0; γ, δ) in section 5.

5 The Definition of the Refined Invariant

5.1

The purpose of this section is to explain how to attach to any triple (γ0; γ, δ)satisfying the conditions in §2 of [9] an invariant α1(γ0; γ, δ). This will be calledthe refined invariant because it will be related to the invariant α(γ0; γ, δ) definedby Kottwitz in [9] in the following way: it will be an element of a certain groupwhich maps naturally to the group k(I0/Q)D to which α(γ0; γ, δ) belongs, andfurthermore α(γ0; γ, δ) will be the image of α1(γ0; γ, δ).

We will recall first the conditions imposed on the triples. A triple (γ0; γ, δ)is supposed to satisfy the following conditions:

1. γ0 ∈ G(Q) is semisimple and R-elliptic.

2. γ = (γl)l ∈ G(Apf ) and for every l 6= p,∞, γl is stably conjugate to γ0.

3. δ ∈ G(Lr) and Nδ = δσ(δ) . . . σr−1(δ) is stably conjugate to γ0.

22

Further we assume that the image of [δ] under the canonical map

B(GQp) −→ X∗(Z(G)Γ(p))

is the negative of the restriction to Z(G)Γ(p) of µ1 ∈ X∗(Z(G)), where µ1 isthe element attached to any h ∈ X∞ as in [9]. It is easy to see that sinceG = GSp(2g) this condition on δ can be rephrased as: c(δ) = p−1u−1, whereu ∈ O×

Lr.

To which group should the refined invariant belong? To answer this writeI0 = Gγ0

and I1 = I0 ∩ Gsc. The reader who is familiar with the stabiliza-tion of the twisted trace formula in [13] might guess that this group shouldbe X∗(Z(I1)

Γ), because the element α1(γ0; γ, δ) should play the same role asobs(δ) of that paper, which for δ regular with centralizer the maximal torusT0 and with T1 = T0 ∩ Gsc, belongs to the group H1(A/Q, T1) ∼= X∗(TΓ

1 )tor.However, we were not able to get a well-defined element of this group from anarbitrary triple (γ0; γ, δ), but only for those for which c(δ) = p−1. This seemsto be due to the fact that we only consider the case of Shimura varieties withgood reduction at p, meaning that in particular Kp is a hyperspecial maximalcompact subgroup of G(Qp). There are several noncanonical ways to produce

an element of X∗(Z(I1)Γ) from a triple (γ0; γ, δ), but probably one would need

to consider also Shimura varieties where Kp is allowed to be an arbitrary para-horic subgroup in order to determine which of these ways is the correct one.Since such varieties are not considered in this paper, we content ourselves toproduce an element of a certain quotient group of X∗(Z(I1)

Γ), and it turns outthat this will be sufficient for the purposes of this paper.

To be precise, we will associate to any triple (γ0; γ, δ) an element α1(γ0; γ, δ)of the group

X∗(Z(I1)Γ)/im(Z×

p ),

and this element will depend on our choice of data D. Here im(Z×p ) denotes the

image of the map

Z×p

// Q×p

∂ // H1(Qp, I1)can // X∗(Z(I1)

Γ(p))torj∗

p// X∗(Z(I1)

Γ).

The second map is the boundary map coming from the exact sequence

1 // I1 // I0c // Gm

// 1,

the third map is the canonical isomorphism extending the Tate-Nakayama iso-morphism for tori that is described in §1 of [8], and the fourth is the obviousmap induced by the field embedding jp in our data D. In a similar way we

can consider the image of Z×p in B(I1), B

p(I1 → I0), X∗(Z(I1)

Γ(p)), and inX(I1 → I0)Qp

, or even X(I1 → I0), where these last two groups are defined inthe obvious way by considering the connected reductive groups I0 and I1 overQp and Q, respectively (see §4.1). We use the same symbol im(Z×

p ) to denoteany of these images. Note that none of them actually depends on the choice of

23

the field embedding jp. Furthermore, in each case the set im(Z×p ) is a subgroup;

this follows easily in the case where I0 and I1 are tori. The general case followsfrom the special case by noting that the image im(Z×

p ) does not change if wereplace I0 with a maximal Q-torus T ⊂ I0 and I1 with T1 = T ∩ I1.

To define α1(γ0; γ, δ), first choose field embeddings jv : Q → Qv for eachplace v of Q. For v = p and v = ∞, use the embeddings already fixed in ourdata D. These give us inclusions Γ(v) → Γ, for every place v. We will firstdefine, for each place v of Q, an element

α1(l) ∈ X∗(Z(I1)Γ(l)), if l 6= p,∞,

α1(p) ∈ X(I1 → I0)Qp/im(Z×

p ),α1(∞) ∈ X(I1 → I0)C.

Here, X(I1 → I0)C = X∗(Z(I0))/A(∞)X∗(Z(I1)), and A(∞) is the augmenta-tion ideal for the group Γ(∞).

Definition of α1(l), for l 6= p,∞: by assumption the elements γ0 and γl areconjugate in G(Ql). Choose an element gl ∈ Gsc(Ql) such that glγ0g

−1l = γl.

Then the 1-cocycle τ → g−1l τ(gl) of Γ(l) with values in I1(Ql) gives us a well-

defined element of H1(Ql, I1). Using the canonical (extension of the) Tate-Nakayama isomorphism mentioned above, we then get an element α1(l) ∈X∗(Z(I1)

Γ(l)). We denote by j∗l (α1(l)) the image (under the obvious map in-

duced by jl) of this element in the group X∗(Z(I1)Γ)/im(Z×

p ). Note that thiselement does not actually depend on the choice of embedding jl.

Next, we define α1(∞). By assumption γ0 is R-elliptic, so we may choose anelliptic maximal R-torus T of G containing γ0. Then T is also a maximal R-torusof I0. We may choose h ∈ X+ which factors through T (R) (use that any twoelliptic maximal R-tori of G are conjugate under Gsc(R)). Let T sc = T ∩ Gsc.It is not hard to see that h is well-defined up to an element of the Weyl groupΩ(Gsc(R), T sc(R)), and thus using Lemma 5.1 of [9], we see that the class ofthe cocharacter µh gives us a well-defined element in X(T sc → T )C. Then wedefine α1(∞) to be the image of [µh] under the restriction map

X(T sc → T )C → X(I1 → I0)C

defined by the canonical inclusions Z(I1) → T sc and Z(I0) → T . This elementdoes not depend on the choices of T and h made in its construction, as one cancheck by using the fact that any two elliptic maximal R-tori in I0 are conjugateunder I1(R). Finally, we denote by j∗∞(α1(∞)) the image of this element in thegroup X(I1 → I0)/im(Z×

p ). The element α1(∞) depends on the embedding j∞,and thus on our data D.

We now define the element α1(p). We have assumed that γ0 and Nδ areconjugate under G(Qp), and that c(δ) = p−1u−1, for some u ∈ O×

Lr. Make a

preliminary choice of d ∈ O×L such that d−1σ(d) = u. Using Steinberg’s theorem

that H1(L, I1) = (1), it is not hard to find a y ∈ G(L) such that

1. yγ0y−1 = Nδ.

24

2. c(y) = d.

By applying σ to the first equation and noting that y is unique up to right trans-lation by an element of I1(L) we see that we get a well-defined (but dependingon d) element [y−1δσ(y)] ∈ Bp(I1 → I0). The purpose of the second conditionon y is to insure that c(y−1δσ(y)) = p−1. It is easy to see what happens tothis element when we change the choice of d. The result is that [y−1δσ(y)] ismultiplied by an element of im(Z×

p ). Therefore, [y−1δσ(y)] is in fact well-defined(independently of our choice of d) in the quotient group Bp(I1 → I0)/im(Z×

p ).Now recall the mapsMg constructed in §4, using the particular element g definedby Reimann and Zink. We define α1(p) to be the element

Mg([y−1δσ(y)]) ∈ X(I1 → I0)Qp

/im(Z×p ).

Further, we denote by j∗p(α1(p)) the image of this element inX(I1 → I0)/im(Z×p ).

The element α1(p) depends on g and thus on the data D.

Definition 5.1.

α1(γ0; γ, δ) =∑

v

j∗vα1(v).

We need to show that we get a well-defined element of X∗(Z(I1)Γ)/im(Z×

p ).At all places l 6= p,∞ it is clear that j∗l α1(l) lies in this group. Moreover, byimitating Proposition 7.1 of [8] it is not hard to prove that almost all of theseelements vanish. Consider the exact sequence

0 // X∗(Z(I1)Γ)/im(Z×

p ) // X(I1 → I0)/im(Z×p )

c // X∗(Gm) // 0

The image of j∗p(Mg(γ−1δσ(y))) under c is −1 (use c(y−1δσ(y)) = p−1). The

image of j∗∞([µh]) under c is 1 (we use µhµ∗h = id). Therefore

j∗p(α1(p)) + j∗∞(α1(∞)) ∈ X∗(Z(I1)Γ)/im(Z×

p ).

This completes the construction of the invariant α1(γ0; γ, δ) depending on thedata D.

Remark 5.2. It is immediate from the definition of the local terms α1(v)above and those going into the definition of Kottwitz’ invariant α(γ0; γ, δ), thatα1(γ0; γ, δ) maps to α(γ0, γ, δ) under the map

X∗(Z(I1)Γ)/im(Z×

p ) → k(I0/Q)D.

5.2 Transformation Laws for α1(γ0; γ, δ).

In this section we will explain the effect on α1(γ0; γ, δ) of changing the triple(γ0; γ, δ).

Proposition 5.3. If γ′0 is stably conjugate to γ0, then

α1(γ′0; γ, δ) = α1(γ0; γ, δ).

25

Proof. Write I ′0 (resp. I0) for the centralizer of γ′0 (resp. γ0). Then the Q-groups I ′1 and I1 (with the obvious meaning) are inner twists of each other,so that we may identify Z(I1) with Z(I ′1), and thus X∗(Z(I1)

Γ)/im(Z×p ) with

X∗(Z(I ′1)Γ)/im(Z×

p ), and it is via this identification that we understand theequality to be proved.

We proceed as in §2 of [9]. Choose g ∈ Gsc(Q) such that gγ0g−1 = γ′0. One

can show that for all v,α′

1(v) = α1(v) − tv.

Here, for v 6= p, tv denotes the image under

H1(Q, I1) → H1(Qv, I1) → X∗(Z(I1)Γ(v))

of the class of the 1-cocycle [τ 7→ g−1τ(g)] of Γ in I1(Q). For v = p, tp is theimage of the same 1-cocycle under the map

H1(Q, I1) → H1(Qp, I1) → X∗(Z(I1)Γ(p))/im(Z×

p ).

Actually, when tracing through the definitions at the place ∞, the reader willfind that α′

1(∞) = α1(∞)+ t∞, but this is equivalent to what we want, becauseX∗(Z(I1)

Γ(∞)) is an elementary abelian 2-group. Now the Proposition followsfrom Proposition 2.6 of [9].

To state the next transformation law, we need to introduce some more no-tation. Suppose that (γ0; γ

′, δ) and (γ0; γ, δ) are two triples such that γl andγ′l are conjugate in G(Ql), for each l 6= p,∞. Then for each l 6= p,∞, we can

find xl ∈ Gsc(Ql) such that xlγlx−1l = γ′l . We can identify Z(Gsc

γl) with Z(I1).

We denote by inv1(γl, γ′l) the image of the 1-cocycle τ 7→ x−1

l τ(xl) under theTate-Nakayama isomorphism (as extended by Kottwitz)

H1(Ql, Gscγl

) → X∗(Z(I1)Γ(l))tor.

Proposition 5.4. For any choice of field embeddings jl, (l 6= p,∞), we have

α1(γ0; γ′, δ) = α1(γ0; γ, δ) +

∑

l 6=p,∞j∗l (inv1(γl, γ

′l)).

Proof. We first note that by the obvious analogue of Proposition 7.1 of [8],almost all of the terms inv1(γl, γ

′l) are trivial, and therefore the sum exists.

Then the result follows from the local result

α′1(l) = inv1(γl, γ

′l) + α1(l),

which is an easy consequence of the definitions, using Lemma 1.4 of [8].

Now we come to the most important transformation laws, which will be directlyused in the proof of the Main Theorem in §8. However, these are only relevant(indeed they can only be stated) for certain triples, namely those triples (γ0; γ, δ)for which γ0 satisfies a certain condition, which we now explain. The element

26

γ0 gives rise to the groups I0 and I1 and, by considering dual groups, to theexact sequence

1 // Z(G) // Z(I0) // Z(I1) // 1,

which induces the boundary map ∂ : Z(I1)Γ → H1(Q, Z(G)). We have fixed

the character ω, which corresponds to the global Langlands parameter a ∈H1(Q, Z(G))/ker1(Q, Z(G)) ∼= H1(Q, Z(G)) (since G is the group of symplecticsimilitudes).

Definition 5.5. We say that γ0 is ω-special if there exists κ0 ∈ Z(I1)Γ such

that ∂(κ0) = a.

Remark 5.6. It is easy to show that if γ ′0 and γ0 are G(Q)-conjugate, thenone is ω-special if and only if the other one is. Therefore it makes sense to calla pair (γ, δ) ω-special if the element γ0 constructed from this pair is ω-special,since γ0 is uniquely determined up to stable conjugacy by (γ, δ).

If γ0 is ω-special with ∂(κ0) = a, then one can show that κ0 satisfies〈im(Z×

p ), κ0〉 = 1 with respect to the canonical pairing

〈·, ·〉 : X∗(Z(I1)Γ) × Z(I1)

Γ −→ C×.

First one proves the equality 〈im(x), κ0〉 = ωp(x), for any x ∈ Q×p , by routine

but tedious diagram chases which relate ω and ωp to the global and local Tate-

Nakayama pairings defined on the torus Iab0 = I0/I

der0 , and thus to κ0. Then the

result follows from the fact that ωp(Z×p ) = 1. Thus it makes sense to consider

the pairing 〈α1(γ0; γ, δ), κ0〉, in the case where γ0 is ω-special. Note that we areusing in an essential way that we are in the case of good reduction here, i.e, Kp

is a hyperspecial maximal compact subgroup and c(Kp) = Z×p .

Theorem 5.7. Suppose that (γ0; γ, δ) is a triple such that γ0 is ω-special, andsuppose that ∂(κ0) = a. Then the following transformation laws hold:

1. If g = (gl)l 6=p,∞ ∈ G(Apf ) and γ′ = gγg−1, then

〈α1(γ0; γ′, δ), κ0〉 = 〈α1(γ0, γ, δ), κ0〉ω(g)−1.

2. If h ∈ G( Lr) and δ′ = hδσ(h)−1, then

〈α1(γ0; γ, δ′), κ0〉 = 〈α1(γ0; γ, δ), κ0〉ω(h)−1.

Proof. (1). Using the preceding Proposition, we are reduced to proving a purelylocal statement:

〈inv1(γl, glγlg−1l ), j∗l (κ0)〉 = ωl(gl)

−1,

where we are using jl to define the inclusion j∗l : Z(I1)Γ → Z(I1)

Γ(l). This iseasy to prove, once again using standard diagram chases to relate j∗l (κ0) to ωl.

27

(2). Here we have not introduced the p-adic analogue of inv1(γl, γ′l), and so we

must argue directly. To define α1(p), write c(δ) = p−1dσ(d)−1, where d ∈ O×L .

Then we can find y ∈ G(L) such that (i) yγ0y−1 = Nδ, and (ii) c(y) = d. Note

that c(δ′) = p−1(c(h)d)σ(c(h)d)−1. We use y to define α1(p) = Mg[y−1δσ(y)].

But we cannot necessarily use the element hy ∈ G(L) (the natural guess!) todefine α′

1(p), because although (i) (hy)γ0(hy)−1 = Nδ′, and (ii) c(hy) = c(h)d,

the element c(h)d ∈ L× might not be in O×L . To remedy this, write c(h) = pnv,

where v ∈ O×Lr

and n ∈ Z. Using Steinberg’s theorem we can find λ ∈ I0(Qunp )

such that c(λ) = p−n. Then it follows easily that we can now use the elementhyλ ∈ G(L) to define α′

1(p), and moreover that we have

α′1(p) = α1(p) +Mg([λ

−1σ(λ)]).

Note that Mg[λ−1σ(λ)] is precisely the image of p−n under the map

Q×p

// H1(Qp, I1) // X∗(Z(I1)Γ(p)) // X(I1 → I0)Qp

/im(Z×p ).

We see then that we are reduced to proving the local statement

〈Mg[λ−1σ(λ)], j∗p(κ0)〉 = ωpc(h)

−1.

which is is just the equality

ωp(p−n) = ωpc(h)

−1.

This in turn follows from c(h) = pnv and the fact that ωp is trivial on O×Lr

.

Remark 5.8. As stated, the equalities above are between complex numbers,since we have regarded the pairing 〈·, ·〉 and the character ω (and thus ω = ω cas having values in C×. However, it is important to note that later in thispaper (especially in the statement of the Main Theorem of §8), we shall be

regarding both sides of the equalities as elements of Q×` . This we do by using

the isomorphism C ∼= Q` that we chose in §2, and which has already been used

there to regard ω as having values in Q×` .

6 Further Reductions

Recall that we are denoting by Fix the set of points [A′, λ′, η′] ∈ SKpg(k) which

are fixed by the correspondence Φrp f . As stated in §2, we want to give a

formula for ∑

[A′,λ′,η′]∈Fix

ω[A′, λ′, η′].

To so so we recall several facts from [11]. In §16 of that paper it is shown thateach fixed point [A′, λ′, η′] gives rise to a positive rational number c = c0p

r (c0a p-adic unit) and a c-polarized virtual abelian variety over kr up to isogeny(A′, λ′). Also, (A′, λ′) automatically satisfies the conditions in §14 of [11], and

28

so gives rise to a triple (γ′0; γ′, δ′), which is well-defined up to an equivalence

relation we will call G-equivalence. (The triples (γ0; γ, δ) and (γ′0; γ′, δ′) are G-

equivalent if γ0 and γ′0 are G(Q)-conjugate, γ and γ′ are conjugate under G(Apf ),

and δ and δ′ are σ-conjugate under G(Qpr ).) Note that (γ′0)(γ′0)

∗ = c−1. So bythese remarks and using the notation from the end of §3, we see that the sumabove can be written as ∑

c

∑

(A,λ)

Tω(A, λ).

The outer sum ranges over all numbers c of the form c0pr, and for each c the

pairs (A, λ) range over those which are c-polarized. We could just as well writeour sum as ∑

c

∑

(γ0;γ,δ)

∑

(A,λ)

Tω(A, λ).

Now c ranges as before. For each c, we then sum over those G-equivalenceclasses (γ0; γ, δ) with γ0γ

∗0 = c−1. Given (γ0; γ, δ) the inner sum is over the

pairs (A, λ) which give rise to a triple which is G-equivalent to (γ0; γ, δ).The first step in simplifying the above sum is to determine which pairs (A, λ)

have Tω(A, λ) 6= 0. To do this fix (A, λ) and set I = Aut(A, λ). Choose βp andβr for (A, λ) and use them to define γ and δ as in §3. Recall that T ω(A, λ) isthen

ω(βpβr)−1

∫

I(Q)\[G(Apf)×G(Lr)]

ω(y)ω(x)fp(y−1γy)φr(x−1δσ(x)) dydx.

Thus we want to know when this integral does not vanish. We choose Haarmeasures on G(Ap

f )γ and Gδσ(Qp) as follows: Choose Haar measures on I(Apf )

and I(Qp). Tate’s Theorem (as generalized in §10 of [11]) and our choice ofβ = (βp, βr) give us isomorphisms

I(Apf )

v−→ G(Apf )γ

I(Qp)v−→ Gδσ(Qp),

which we use to transport the measures to the right hand side. Since all thesegroups are unimodular, the measures on the right do not depend on the choiceof β.

We also want to transport the character ω to the groups I(Apf ), I(Qp),

and I(A). We know that I(Q) is discrete in I(Af ) and has finite co-volume.Note also that ω is already defined on the groups on the right (it is defined onGδσ(Qp) because Gδσ , Qp

is an inner form of Gγ0 , Qp). Therefore we can use

the isomorphisms to define ω on I(Af ) (and it does not depend on the choiceof β). We could also transport the character ω to the group I(A) using globalLanglands parameters. Namely, the restriction of ω to I0(A) (where I0 = Gγ0

)comes from a global Langlands parameter for the group I0. Since the Q-groupsI0 and I are inner forms, we may identify the centers of their dual groups andvia this identification we get a global Langlands parameter for I, and thus a

29

character on I(A). On I(Af ) this agrees with the one constructed above, butnow we have a character at the infinite place as well as the finite ones. However,since the involution ∗ on End(VR) is positive, the characters on I(R) and I0(R)are both trivial anyway.We need some more definitions.

Definition 6.1. If ω(G(Apf )γ) = 1, let Oω

γ (fp) denote the integral

∫

G(Apf)γ\G(Ap

f)

ω c(y) fp(y−1γy) dy,

where dy is determined by the measure on G(Apf ) giving Kp measure 1 and the

measure on G(Apf )γ specified above. Also, fp is the characteristic function for

Kpg−1Kp.

Definition 6.2. If ω(G(Apf )γ) = 1, let Oω

γ (fp) denote the integral above with

fp replaced by fp, the characteristic function for Kpg−1, and dy replaced withthe measure giving Kp

g = Kp ∩ gKpg−1 measure 1.

Remark. It is straightforward to check that

Oωγ (fp) = Oω

γ (fp)

(Remember ω c(Kp) = 1.)

Definition 6.3. If ω(Gδσ(Qp)) = 1, let TOωδ (φr) denote the integral

∫

Gδσ(Qp)\G(Lr)

ω c(x)φr

(x−1δσ(x)

)dx,

where dx is determined by the measure on G(Lr) giving Kr measure 1 and themeasure on Gδσ(Qp) specified above. Here c(x) is the image of c(x) ∈ L×

r /O×Lr

in Q×p /Z

×p , and φr is the characteristic function for KraKr, where a = µh(p−1)

for any h ∈ X∞, as defined in §3.

Remark. If we replace φr in the integral with φr from §3, then one can checkthat

TOωδ (φr) = TOω

σ−1δ(φr) = ω c(δ)TOωδ (φr).

These equalities are straightforward to prove when one keeps in mind that forall x ∈ G(Lr), ω c(x) = ω c(σ(x)). Also, since the triples (γ0; γ, δ) we areconsidering have c(δ) = p−1u−1 for some u ∈ O×

Lr, we can write ω c(δ) =

ωp(p−1), ωp being the p-component of ω. So the last term above can be written

ωp(p−1)TOω

δ (φr).

For future reference we record here the transformation laws for these (twisted)orbital integrals.

30

Lemma 6.4. For g ∈ G(Apf ) and h ∈ G(Lr) we have

Oωgγg−1(fp) = ω(g) Oω

γ (fp)

TOωhδσ(h−1)(φr) = ω(h) TOω

δ (φr)

Proof. Straightforward computations. Note also that for each equation, the lefthand side is defined if and only if the right hand side is.

We can now finish the first step by proving the following result.

Proposition 6.5. Tω(A, λ) = 0 unless ω(I(Af )) = 1, in which case Tω(A, λ)is given by

vol(I(Q)\I(Af )) ωp(p−1) ω(βpβr)

−1 Oωγ (fp) TOω

δ (φr).

Proof. Suppose J ⊆ H ⊆ G are three unimodular groups. We have the integra-tion formula ∫

J\G

φ(g) dg =

∫

H\G

(∫

J\H

φ(hg) dh

)dg.

If we apply this to the integral expression for T ω(A, λ) (take J = I(Q), H =G(Ap

f )γ × Gδσ(Qp), G = G(Apf ) × G(Lr)), then the inner integral contains the

factor ∫

I(Q)\[G(Apf)γ×Gδσ(Qp)]

ω(hpy) ω(hpx) dhp × dhp,

which is 0 unless ω(G(Apf )γ × Gδσ(Qp)) = 1 (that is, ω(I(Af )) = 1), in which

case it isvol(I(Q)\I(Af )) ω(y)ω(x).

Now considering the outer integral shows that in this case T ω(A, λ) is

vol(I(Q)\I(Af )) ω(βpβr)−1 Oω

γ (fp) TOωδ (φr),

and the proposition is a consequence of this and the preceding remarks.

The second step in simplifying our sum is to fix a (A, λ) such that T ω(A, λ) 6=0, let I, βp, βr and (γ0; γ, δ) be as above, and consider the part of the sum over all(A′, λ′) which give rise to a triple G-equivalent to (γ0; γ, δ). Let I ′ = Aut(A′, λ′).Let I0 = Gγ0

.

Lemma 6.6.

ω(I0(A)) = ω(I(A)) = ω(I ′(A)) = 1.

Proof. By §14 of [11] we know that there is an inner twisting over Q

I0v−→ I

which is unique up to inner automorphisms of I0(Q). Therefore I is a connectedreductive Q-group because I0 is, by a theorem of Steinberg. It follows that I(Q)is dense in I(R), and so

I(A) = I(Q)[I(Af )I(R)0].

31

Since ω(I(Af )) = 1, we see from this that ω(I(A)) = 1. (Alternatively, we canavoid invoking the real approximation theorem for this point by using the factthat the involution ∗ on End(VR) is positive.) The groups I0 and I ′ are eachinner forms of I, and so ω(I ′(A)) = ω(I0(A)) = ω(I(A)) = 1.

Remark. This type of result is easy to see directly if we assume that γ0 is aregular semisimple element. In that case I0, I, and I ′ are all tori, isomorphicover Q. Moreover, we note that in this case the Q-isomorphism I0

v−→ I isunique, and over Ql for any l 6= p, ∞ it is the isomorphism constructed byusing Tate’s theorem, our choice of βl, and the fact that γ0 and γl are stablyconjugate. A similar argument works for the finite place p and that is sufficientbecause the characters are automatically trivial at ∞.

Let c2(γ0; γ, δ) = vol(I(Q)\I(Af )), and c1(γ0) = |ker1(Q, I)|. Let c(γ0; γ, δ) =c1(γ0)c2(γ0; γ, δ). Note that these numbers really do depend only on (γ0; γ, δ)and remain unchanged if we replace I by I ′. The set of pairs (A′, λ′) we areconsidering is in bijective correspondence with ker1(Q, I), by Theorem 17.2 of[11]. We describe this explicitly: Any Q-isogeny θ : (A′, λ′) → (A, λ) gives anelement [θ τ(θ−1)] ∈ H1(Q, I), which lies in ker1(Q, I) if and only if (A′, λ′)gives a triple G-equivalent to (γ0; γ, δ). If this element is in ker1(Q, I), choosea finite Galois extension K/Q such that [θ τ(θ−1)] ∈ H1(K/Q, I). Then thereexists an element

ψ = (ψv)v ∈∏

v

I(Qv ⊗K),

such that

1. ψv ∈ I(Zv ⊗K) for almost every finite place v of Q, and

2. θ τ(θ−1) = ψ−1v τ(ψv), ∀ τ ∈ Γ(v), for every place v.

Then it follows that we can use βpψpθ and βrψpθ to define γ′ and δ′ for (A′, λ′).When we do so we find that γ′ = γ and δ = δ′. We thus see that

Tω(A′, λ′) = ω(ψpθ)−1ω(ψpθ)−1Tω(A, λ).

Now recall a ∈ H1(Q, Z(G))/ker1(Q, Z(G)), the global Langlands parametercorresponding to ω : G(A) → C×. Let I1 = I0 ∩ Gsc. We have the exactsequence

1 //Z(G) //Z(I0) //Z(I1) //1

which gives rise to maps (see §6.2 of [13])

α : H1(Q, Z(G))/ker1(Q, Z(G)) −→ H1(Q, Z(I0))/ker1(Q, Z(I0)),

β : kerα −→ coker[ker1(Q, Z(G)) → ker1(Q, Z(I0))].

Note that in our situation ker1(Q, Z(G)) = (1), since G is the group of sym-plectic similitudes. Because ω(I0(A)) = 1 we know from the theory of globalLanglands parameters that a ∈ kerα, and thus we get an element β(a) ∈

32

ker1(Q, Z(I0)). We denote by inv((A′, λ′), (A, λ)) the element [θ · τ(θ−1)] inker1(Q, I0) (here we can identify ker1(Q, I0) with ker1(Q, I), since I and I0 areinner forms).

Proposition 6.7.

〈inv ((A′, λ′), (A, λ)), β(a)〉 = ω(ψpθ) ω(ψpθ),

where〈·, ·〉 : ker1(Q, I0) × ker1(Q, Z(I0)) −→ C×

is the Tate-Nakayama pairing (as extended by Kottwitz) on these finite abeliangroups (see §4 of [6]).

Proof. This involves a straightforward unwinding of the definition of the afore-mentioned Tate-Nakayama pairing. It is very similar to the argument given in§6.2 of [13], where the required diagram chase is carried out in a much moregeneral situation. We omit the details.

Now we can complete the second step of the simplification.

Corollary 6.8. The sum of Tω(A′, λ′) over all pairs (A′, λ′) which give riseto the G-equivalence class of (γ0; γ, δ) is zero unless ω(I0(A)) = 1 and β(a) ∈ker1(Q, Z(I0)) is trivial. When these conditions hold, the sum is given by

c(γ0; γ, δ) ωp(p−1) ω(βpβr)

−1 Oωγ (fp) TOω

δ (φr).

Proof. If ω(I0(A)) 6= 1 then we have already seen that each term T ω(A′, λ′) = 0.Suppose ω(I0(A)) = 1, so that β(a) exists. As (A′, λ′) varies, inv((A′, λ′), (A, λ))ranges over the elements x of ker1(Q, I0). So using the preceeding two Proposi-tions, our assertion follows from the fact that

∑

x

〈x, β(a)〉

is 0 if β(a) is nontrivial, and is |ker1(Q, I0)| if β(a) is trivial.

Now recalling Theorem 18.1 of [11] and imitating the reasoning in §19 of thatpaper, we see that our sum

∑

c

∑

(γ0;γ,δ)

∑

(A,λ)

Tω(A, λ)

can now be written as∑

(γ0;γ,δ)

c(γ0; γ, δ) ωp(p−1) ω(βpβr)

−1 Oωγ (fp) TOω

δ (φr),

where (γ0; γ, δ) ranges over G-equivalence classes of triples such that

1. The Kottwitz invariant α(γ0; γ, δ) is trivial,

33

2. ω(I0(A)) = 1,

3. β(a) is trivial,

and where, for any pair (A, λ) giving rise to the G-equivalence class of (γ0; γ, δ),βp and βr are chosen for (A, λ) and are used to get γ and δ. Note that if thesummand indexed by (γ0; γ, δ) in this last sum is nonzero, then there is some cand a c-polarized virtual abelian variety over kr up to isogeny (A, λ) that givesrise to the G-equivalence class of (γ0; γ, δ). Moreover, this summand does notthen depend upon the choice of (A, λ), βp, βr giving rise to the triple. Thereforethe notation makes sense.

Lemma 6.9. Conditions 2. and 3. above hold if and only if γ0 is ω-special.

Proof. This is an easy consequence of the definitions.

By the Lemma, we see that the index set of the second sum is precisely the setof G-equivalence classes of triples (γ0; γ, δ) such that

1. The Kottwitz invariant α(γ0; γ, δ) is trivial,

2. γ0 is ω-special.

Note that ω(βpβr)−1 is the only term appearing in the summand indexed by

(γ0; γ, δ) that is not purely group-theoretic in nature: one needs to make refer-ence to an abelian variety to define this number. It turns out that the refinedinvariant α1(γ0; γ, δ) is precisely what is needed in order to remedy this:

Theorem 6.10. Let (A, λ) be a c-polarized virtual abelian variety over kr upto isogeny, and suppose that βp and βr are symplectic similitudes used to defineγ and δ. Suppose that (γ, δ) is ω-special, and that κ0 ∈ Z(I1)

Γ is any elementwith ∂(κ0) = a. Then

〈α1(γ0; γ, δ), κ0〉 = ω(βpβr)−1.

Note that we can regard this as an equality of elements in either C or Q`.The preceding simplication of our sum and this theorem immediately implythe Main Theorem of §8. But before we prove this theorem we must use thetheorem of Reimann and Zink to prove a result about abelian varieties withcomplex multiplication. This is the crucial step, and is done in the next section.

7 Study of an Invariant Attached to Abelian Va-

rieties

In order to prove the necessary properties of α1(γ0; γ, δ), we need to introduceand study another invariant which is attached to a polarized abelian variety overk with an action by a CM-algebra. We will in fact prove that this invariant isalways trivial, and this will turn out to yield key information about α1(γ0; γ, δ).

34

Let F be a CM-algebra with involution ∗. By this we mean F is a productof totally imaginary quadratic extensions of totally real number fields, and ∗induces complex conjugation on each factor of F ⊗ R ∼= C × · · · × C. Let(V, 〈·, ·〉) be a symplectic vector space over Q with CM by F . This means thatV is a free F -module of rank 1 and the relation 〈xv,w〉 = 〈v, x∗w〉 holds forx ∈ F and v, w ∈ V . Let T be the Q-torus x ∈ F× | xx∗ ∈ Q×, and let T1

be the Q-torus x ∈ F× | xx∗ = 1. Recall from §13 of [11] that a morphismf : (V1; 〈·, ·〉1) → (V2; 〈·, ·〉2) between two symplectic vector spaces with CM byF is by definition an F -linear map such that f ∗〈·, ·〉2 = c〈·, ·〉1 for some c ∈ Q×.We call such a morphism strict if c = 1. Therefore we have

T1 = Autstrict(V, 〈·, ·〉),T = Aut(V, 〈·, ·〉).

When we want to regard (V, 〈·, ·〉) as an object in the category with strict mor-phisms, we call it a strict symplectic space.

Let A be an abelian variety over k up to Q-isogeny, and suppose that thereis an injection i : F → End(A). Further suppose that 2 dim(A) = [F : Q], andthat there is a Q-polarization λ : A → A such that i is a ∗-homomorphism forthe involution on F and the Rosati involution ıλ on End(A). For such a pair(A, λ) we will define an invariant α(A, λ) ∈ X∗(T1)Γ.

To do this we choose embeddings jv : Q → Qv for every place v; for v = p,∞use the embeddings in our data D. These allow us to regard each local Galoisgroup as a subgroup of the global Galois group: Γ(v) → Γ. Using D we get anelement g as in §1 of [27] and §4.2 above. Thus we get an isomorphism

K−1g : Bp(T1 → T ) −→ X(T1 → T )

from the exact sequence

1 // T1// T

x7→x∗x// Gm// 1.

(Recall that NmL(√

p)/L(g) = p and that we have proved in §4.2 that KΠ isindependent of the choice of tower Π over g for tori of this form; this justifiesthe notation Kg.) To define α(A, λ) we will define α(v) ∈ X∗(T1)Γ(v) for eachplace v:

For l 6= p,∞, note that the l-adic Tate module H1(A,Ql) is a strict symplec-tic space with CM by F (the pairing comes from the polarization λ, and we viewthe Ql(1)(k)-valued pairing as Ql-valued using D as in §1). Note that H1(A,Ql)and V ⊗Ql become isomorphic over Ql as strict symplectic spaces with CM byF . Therefore their difference is measured by an element of H1(Ql, T1). Nowusing the Tate-Nakayama isomorphism

H1(Ql, T1)v−→ X∗(T1)Γ(l),tor ,

we get an element α(l) of X∗(T1)Γ(l). Using

j∗l : X∗(T1)Γ(l) −→ X∗(T1)Γ

35

we get an element j∗l (α(l)) ∈ X∗(T1)Γ.Now consider the place p. Let (D,Φ) denote the dual of the L-isocrystal

(H1crys(A/W (k)) ⊗ L,Fr). Then D is a strict symplectic space with CM by

F , and Φ commutes with the action of F . Note that D and V become strictlyisomorphic over L and so by Steinberg’s Theorem that H1(L, T1) = (1), theybecome isomorphic over L. Choose a strict isomorphism

Dv−→ V ⊗ L

and use it to transport the σ-linear bijection Φ to bσ on the right hand side. Thusb ∈ T (L) and bb∗ = p−1. Changing the choice of the above strict isomorphismonly changes b by an element of A〈σ〉T1(L), so we have a well-defined element[b] ∈ Bp(T1 → T ). We define α(p) = K−1

g ([b]), an element of X(T1 → T )Qp.

Then j∗p(α(p)) is an element of X(T1 → T )Q.Finally at the place ∞ we let

h : C → EndF (VR) = F ⊗ R

be the unique ∗-homomorphism such that 〈·, h(i)·〉 is positive definite on VR×VR.Then we get a cocharacter µh ∈ X∗(TC). We define α(∞) to be the class of µh

in X(T1 → T )C. Thus j∗∞(α(∞)) ∈ X(T1 → T )Q.

Definition 7.1.

α(A, λ) =∑

v

j∗v (α(v)).

Remark. We have to show this gives us a well-defined element of X∗(T1)Γ. Forv 6= p,∞, it is clear that j∗v (α(v)) lies in this group. Moreover, it is not toodifficult to show that for almost all v, j∗v (α(v)) = 0. Recall that we have theexact sequence

0 // X∗(T1)Γ // X(T1 → T )Qc // X∗(Gm) // 0

where we use c for the map x 7→ x∗x. Tracing through the definitions showsthat µ∗

hµh = id, so that the image of j∗∞(α(∞)) under c is 1. The image ofj∗p(α(p)) under c is −1 since K−1

g [p−1] = −1. Therefore

j∗p(α(p)) + j∗∞(α(∞)) ∈ X∗(T1)Γ.

It still appears that α(A, λ) depends on the choices of the field embeddingsjv. The following result shows that we do not need to worry about this.

Theorem 7.2. α(A, λ) = 0.

Proof. The strategy is similar to that of the proof of Lemma 13.2 in [11]. Thefirst step is to notice that α(A, λ) is independent of the choice of symplectic space(V, 〈·, ·〉) with CM by F we made at the outset. This is proved by imitating theproof of the analogous step in Lemma 13.2 of [11], but since we are concernedwith strict symplectic spaces we use the torus T1 instead of T throughout (see

36

also the proof of Proposition 5.4). Next we can verify that α(A, λ) is independentof the choice of polarization λ. Again this is tedious but not hard. Using thesame arguments, we can show that α(A, λ) is not changed if we replace A byan abelian variety in the same Q-isogeny class. Now by a theorem of Tate [32],A is Q-isogenous to the geometric special fiber of an abelian scheme A1 overOK with CM by F , OK being the integers of some p-adic field K, which we canchoose large enough so that |Hom

Q−alg(F,K)| = [F : Q]; thus K splits TQp.

Moreover, there is a Q-polarization λ1 of the F -abelian scheme A1/OK , by theargument of §5 in [11], where the valuative criterion of properness is verified forcertain Shimura varieties.

Therefore for the purposes of proving the theorem we may assume (A, λ) isthe reduction modulo p of (A1, λ1). We change notation and write (A, λ) for theabelian variety over OK and (A, λ) for the geometric special fiber. We choosej : Qp

v−→ C such that j jp = j∞, and by base changing via j we can considerthe abelian variety over C, which we denote by AC. The polarization λ can alsobe regarded over C in this way. By our remarks above, in defining α(A, λ) weare free to use the strict symplectic space

(H1(AC,Q), − Eλ(·, ·)),

where Eλ(·, ·) denotes the Riemann form attached to λ, as in [22] for example.

Claim 7.3.

α(l) = 0 ∀ l 6= p,∞.

Proof. We have isomorphisms of strict symplectic spaces with CM by F :

H1(A,Ql) = H1(AQp,Ql) = H1(AC,Ql).

The first equality holds because the base change isomorphisms for etale(co-)homology preserve the first Chern classes of line bundles, and therefore pre-serve the Ql(1)-valued pairings (·, ·)λ on both sides. (Here we identify Ql(1)(k)with Ql(1)(Qp) via reduction modulo p : Zun

p → k.) For the second equalitywe consider the pairing on the left hand side to be Ql-valued by first using j toidentify Ql(1)(Qp) with Ql(1)(C) and then by using exp(2πix) to identify thisgroup with Ql (the same i ∈ C as always). Now the theorem on p.237 of [22]shows that this pairing is precisely the pairing −Eλ(·, ·) on the right hand side,and thus the second equality holds. This proves the claim.

Now we need to describe α(p). Choose an isomorphism of strict symplecticspaces with CM by F over L:

(D, (·, ·)λ)f−→ (H1(AC, L),−Eλ(·, ·)). (1)

This is what we use to define α(p), and it must be compared to the situationReimann and Zink consider in [27]. To do this consider the p-adic Tate moduleTp(A) = H1(AQp

,Zp). Reimann and Zink choose a strict isomorphism

Dv−→ Tp(A) ⊗ L (2)

37

where the right hand side has L-valued pairing by using j and exp(2πix) toidentify Qp(1)(Qp) with Qp. They define b′ ∈ T (L) by the equality

Φ−1 = σ−1b′.

Now recall that the action of F on Lie(A) (over K) provides a cocharacterµCM ∈ X∗(TQp

). (Use j and LieAC ⊕ LieAC = H1(AC,Q) ⊗ C to prove theexistence and the compatibility of µCM with the complex structure on Lie(AC).)The theorem of Reimann and Zink (Theorem 4.13) says that if i, jp and j∞ areused in defining g, then

Kg[µCM ] = [b′],

where [b′] is the class of b′ in Bp(T1 → T ). But now again by [22], p.237 we notethat we can identify the right hand side of (1) and (2) above, and so in definingα(p) we may use the map in (2). Thus we see that α(p) = K−1

g [b′]−1, so thatthe theorem of Reimann and Zink can be interpreted as α(p) = −[µCM ].

Finally, we need a description of α(∞). We have to find the unique ∗-homomorphism h : C → EndF (VR) such that 〈·, h(i)·〉 is positive definite.

Claim 7.4. The desired h is given by the complex structure on Lie(AC) = VR.

Proof. Recall that 〈·, ·〉 = −Eλ(·, ·). The relation −Eλ(iv, iw) = −Eλ(v, w)shows that h is a ∗-homomorphism. It is obvious that the F -action commuteswith the complex structure, so h(i) is an element of EndFVR. Finally, the firstchapter of [22] shows that Eλ(i·, ·) is positive definite on VR, and this showsthat 〈·, h(i)·〉 = −Eλ(·, i·) is positive definite. This proves the claim.

The aforementioned compatibility of µCM with the complex structure on Lie(AC)implies that j∗(µh) = µCM . Now α(A, λ) = 0 follows from the equalities

j∗p α(p) + j∗∞α(∞) = −j∗p [µCM ] + j∗∞[µh]

= −j∗pj∗[µh] + j∗∞[µh]

= 0.

8 Proof of the Main Theorem

In this section we give the proof of Theorem 6.10. The first step is to showthat both sides of the desired equality are trivial when βp and βr are chosenin a particular way. First we need a preliminary construction. Suppose that(A, λ) and the pair (βp, βr) are used to define the elements γ, δ, and thus alsothe element γ0 (which is only determined up to Q-conjugacy). As before, writeI0 = Gγ0

, and I = Aut(A, λ). As in §14 of [11], write M = End(A) andM0 = Endγ0

(V ), and consider the functor Ψ of Q-algebras to sets where foreach Q-algebra R, Ψ(R) is the set of all ∗-isomorphisms ψ : M0 → M over Rtaking γ0 to π−1

A . We know from §14 of [11] that Ψ(Q) is nonempty, and any

38

two of its elements differ from each other by an inner automorphism of I0(Q).Moreover any element of Ψ(Q) restricts to an inner twisting of Q-groups I0 → I.Choose ψ ∈ Ψ(Q), and write ψv = ψ ⊗ 1 ∈ Ψ(Qv) for the map resulting fromψ by extension of scalars from Q to Qv, for each place v of Q. For every finiteplace v of Q, our choice of βv and Tate’s theorem gives us another elementψ(v) ∈ Ψ(Qv), and moreover the elements ψv and ψ(v) differ from each otherby an inner automorphism of I0(Qv).

Now choose a maximal Q-torus T of I which is elliptic at the infinite placeand at all finite places of Q where I0 is not quasisplit (the existence of T followsfrom the existence of the local tori, together with weak approximation for thevariety of maximal tori in I; see Cor. 3, p.405 of [25]). The torus T transfersto I0 locally everywhere. Further, by the argument of §14 of [11], we see thatthe obstruction to T tranferring to I0 globally vanishes, because T is ellipticat the infinite place; hence T transfers to I0 globally. Therefore the element[ψ τ(ψ)−1] of H1(Q, Iad) which defines the inner twist I0 of I is the imageof an element [tτ ] ∈ H1(Q, Tad), where Tad denotes the image of T in Iad.Therefore by making a different choice of ψ we can assume

Int(tτ ) = ψ τ(ψ)−1,

for every τ ∈ Γ. Now let N = CentMT . Then N is a maximal commutativesemisimple subalgebra of M , and is free of rank two over N0 = x ∈ N |x∗ = x.It follows that N is a CM-algebra of rank 2(dimA), in the sense discussedearlier. The equation above implies that φ = ψ−1 : M → M0 restricts to givean embedding φ : N →M0, which is defined over Q. We also get a Q-embeddingφ : T → I0, because

T (R) = x ∈ NR | x∗x ∈ R×.

We have just constructed a Q-torus inside I0 which is the group of automor-phisms of the CM-algebra F = φ(N). From now on write φ(T ) = T0 andT1 = T0 ∩ Gsc. This torus T0 is used in the proof of the next proposition: ifγ0 were regular, the proof of this proposition would be simpler because then I0would itself be the torus of automorphisms of a CM-algebra. We need the torusT0 in the general case.

Proposition 8.1. Let c = c0pr be a positive rational number, with c0 a p-adic

unit. Let d ∈ O×L be such that d−1σr(d) = c0. Let (A, λ) be a c-polarized virtual

abelian variety over kr up to isogeny which satisfies the conditions in §14 of[11]. Suppose

βp : (H1(A,Apf ) , (·, ·)λ) −→ (V ⊗ A

pf , 〈·, ·〉)

βr : (H(A) , d(·, ·)λ) −→ (V ⊗ Lr , 〈·, ·〉)are symplectic isomorphisms. Suppose that we use βp and βr to define γ and δ.Then

α1(γ0; γ, δ) = 0.

39

Proof. Note that the statement to be proved is independent of the choice of γ0,by Proposition 5.3. Therefore we fix one choice of γ0 and define I0, M0 and thefunctor Ψ as above. This yields T , N , φ, F , T0, T1 just as before. It is clearthat (V, 〈·, ·〉) is a symplectic space with CM by F and that

T1 = Autstrict(V, 〈·, ·〉),T0 = Aut(V, 〈·, ·〉).

We see that A has CM by the CM-algebra F (action is through that of N via

φ−1), and furthermore that the Q-polarization λ : A → A commutes with theaction of F . Therefore we can define α(A, λ) ∈ X∗(TΓ

1 ), as in §7. Since wehave proved in Theorem 7.2 that this element is always trivial, to prove thatα1(γ0; γ, δ) is trivial it suffices to show that for every place v of Q, α(v) 7→ α1(v)under the maps

X∗(TΓ(l)1 ) −→ X∗(Z(I1)

Γ(l)), l 6= p,∞,X(T1 → T0)Qp

−→ X(I1 → I0)Qp/im(Z×

p ),X(T1 → T0)C −→ X(I1 → I0)C

induced by the canonical inclusions Z(I1) → T1 and Z(I0) → T0. Recall thatwe have chosen in our preliminary construction an element ψ ∈ Ψ(Q). Writeφ = ψ−1, φv = ψ−1

v , and φ(v) = ψ(v)−1. Then for every finite place v thereexists hv ∈ I1(Qv) such that Int(hv) φ(v) = φv. We will use these elements hv

to prove that α(v) 7→ α1(v).First consider the case l 6= p,∞. Then to define α1(l) we chose gl ∈ Gsc(Ql)

such that glγ0g−1l = γl. Thus to define the element α(l) we may use the map

H1(A,Ql)hlg

−1

lβl

// V ⊗ Ql.

To see this, note that this map is a symplectic isomorphism (as βl is), andpreserves the CM-algebra actions because N is transported by this map over to

Int(hl)φ(l)(N) = φl(N) = F.

Since hl ∈ I1(Ql), it follows that the 1-cocycle [hlg−1l βlτ(hlg

−1l βl)

−1] maps to[g−1

l τ(gl)] underH1(Ql, T1) −→ H1(Ql, I1),

and thus via the Tate-Nakayama isomorphisms we see that α(l) 7→ α1(l).For the case v = ∞, recall that α1(∞) was defined by choosing an elliptic

maximal R-torus T ′ of G containing γ0 and an h′ ∈ X+ such that h′ factorsthrough T ′(R). Then we defined α1(∞) using µh′ . On the other hand, α(∞) isdefined using µh, where

h : C −→ EndF (V ⊗ R) = F ⊗ R

is the unique ∗-homomorphism such that 〈·, h(i)·〉 is positive definite on V ⊗R.It follows that we may take T ′ = T0 and h′ = h, and so α(∞) 7→ α1(∞).

40

Finally, consider the case v = p. To define α1(p), recall that we choosey ∈ G(L) such that (i) yγ0y

−1 = Nδ = γp, and (ii) c(y) = d. (We can use dhere because c(βr) = d ⇒ c(δ) = p−1dσ(d)−1.) Then α1(p) = Mg[y

−1δσ(y)].Now consider the set

g ∈ I1 | Int(g) φ(p) = φp ,

where we are considering equality only as maps T → I0. This set is a left T1-torsor, defined over L. Moreover, it has a point over L, namely the element hp

constructed above. Steinberg’s Theorem then implies that this set has a pointh over L. Now we can define α(p) using the map

H(A)hy−1βr// V ⊗ L.

To see this, note that this map preserves pairings, since c(y) = d = c(βr). Also,by our choice of h, T is transported over to

Int(h)φ(p)(T ) = φp(T ) = T0.

Therefore N is transported over to CentM0T0 = F . Finally, note that Φ is trans-

ported over to [hy−1δσ(y)σ(h−1)σ], and thus α(p) = Mg[hy−1δσ(y)σ(h−1)].

From this it is obvious that α(p) 7→ α1(p).

We can now give the Proof of Theorem 6.10:

Proof. Choose d ∈ O×L such that d−1σr(d) = c0. We first note that because we

are assuming that the c-polarized virtual abelian variety (A, λ) comes from afixed point of our correspondence, there exist maps

βp : (H1(A,Apf ), (·, ·)λ) −→ (V ⊗ A

pf , 〈·, ·〉)

βr : (H(A), d(·, ·)λ) −→ (V ⊗ Lr, 〈·, ·〉),for which (βp)∗〈·, ·〉 = (·, ·)λ and β∗

r 〈·, ·〉 = d(·, ·)λ. For βp this is a con-sequence of the existence of some symplectic similtude βp (because there is alevel structure η) and the fact that c : G(Ap

f ) → (Apf )× is surjective. For βr this

follows from Lemma 7.2 of [11]. Let γ and δ be constructed using βp and βr. Itis obvious that in this case the right hand side of the equation in Theorem 6.10is trivial, and we have just proved in Proposition 8.1 that the left hand side isalso trivial. Thus the conclusion of Theorem 6.10 holds in this special case. Thegeneral case (where βp and βr are arbitrary symplectic similitudes) follows fromthe special case and the fact that both sides of the equation in Theorem 6.10transform in the same way. Indeed, if we take g ∈ G(Ap

f ) (resp. h ∈ G(Lr))and replace a symplectic isomorphism βp with gβp (resp. replace a symplecticisomorphism βr with hβr), then the result is that both sides of the equation inTheorem 6.10 are

ωc(g)−1ωc(h)−1,

because of Theorem 5.7.

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We conclude with statement of the Main Theorem, which has now been proved:

Theorem 8.2. Let G = GSp2g. Suppose p is an odd prime. Let Kp ⊂ G(Qp) bea hyperspecial maximal compact subgroup, and let Kp ⊂ G(Ap

f ) be a sufficientlysmall compact open subgroup, as in §1. Let SK denote the Shimura varietyassociated to K = KpKp as in §1. Let f denote the Hecke correspondence

coming from an element g ∈ G(Apf ). Fix a character ω : πp → Q

×` (the group

πp is identified with the set of connected components of SK as in §2). Set

ω = ω c : G(A) → Q×` . Let L(ω) denote the operator on H i

c(SK⊗Q , Q`) as in§2. Then the virtual trace

Tr(Φrp f L(ω) ; H•

c (SK⊗Q ,Q`))

is zero unless ω(g) = 1, in which case there exists a natural number r(f) suchthat for all r ≥ r(f) the virtual trace is given by the expression

∑

(γ0;γ,δ)

c(γ0; γ, δ) ωp(p−1) 〈α1(γ0; γ, δ), κ0〉 Oω

γ (fp) TOωδ (φr),

where we take the sum over all G-equivalence classes of triples (γ0; γ, δ) suchthat

1. α(γ0; γ, δ) = 0,

2. γ0 is ω-special,

and where κ0 ∈ Z(I1)Γ is any element which satisfies ∂(κ0) = a.

Here a ∈ H1(Q, Z(G)) is the Langlands parameter corresponding to thecharacter ω, and ∂ : Z(I1)

Γ → H1(Q, Z(G)) is the boundary map arising fromthe exact sequence of Γ-modules

1 //Z(G) //Z(I0) //Z(I1) //1,

where I0 = Gγ0and I1 = Gsc

γ0.

Remark 8.3. Recall that the invariant α1(γ0; γ, δ) was constructed only afterwe fixed a choice of initial data D. This is not too disturbing, however, becausethe operators L(ω) on H i

c(SK ⊗k , Q`) which we used (via the base changetheorems of etale cohomology) to find an expression for the virtual trace, arealso defined only after choosing D, and the dependence of α1(γ0; γ, δ) on Dmerely reflects this fact.

Acknowledgements: This paper is my University of Chicago Ph.D. thesis. Iwould like to thank most of all my thesis supervisor Robert Kottwitz, for allhis help and encouragement over the years, and for suggesting the problem. Iwould also like to thank Matthias Pfau for sending me an unpublished paper ofhis, which provided inspiration for the results in section 3. Finally, I would liketo thank the Max-Planck-Institut fur Mathematik in Bonn, where this paperwas written into its current form, for support and hospitality.

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University of TorontoDepartment of Mathematics100 St. George StreetToronto, ON M5S 1A1, CANADAemail: haines@math.toronto.edu

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